fundamental theorem of calculus formula

Stokes' theorem is a vast generalization of this theorem in the following sense. The derivative of a function is defined as y = f (x) of a variable x, which is the measure of the rate of change of a variable y changes with respect to the change of variable x. dx is the integrating agent. Everyday financial problems such as calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy. Then, separate the numerator terms by writing each one over the denominator: \[ ∫^9_1\frac{x−1}{x^{1/2}}\,dx=∫^9_1 \left(\frac{x}{x^{1/2}}−\frac{1}{x^{1/2}} \right)\,dx. If f is a continuous function and c is any constant, then f has a unique antiderivative A that satisfies A(c) = 0, and … We can use the relationship between differentiation and integration outlined in the Fundamental Theorem of Calculus to compute … Thus, the two parts of the fundamental theorem of calculus say that differentiation and integration are inverse processes. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The first thing to notice is that the Fundamental Theorem of Calculus requires the lower limit to be a constant and the upper limit to be the variable. Suppose f is continuous on an interval I. The Fundamental Theorem of Calculus is the formula that relates the derivative to the integral Let’s double check that this satisfies Part 1 of the FTC. The second part of the theorem gives an indefinite integral of a function. The Fundamental Theorem of Calculus This theorem bridges the antiderivative concept with the area problem. Well the formula in my pdf file where i'm learning calculus is d/dx(integral f(t)dt) = f(x) But i don't seem to graps this formula very well, what does it exactly mean in … limit and is also useful for numerical computation. So, for convenience, we chose the antiderivative with \(C=0\). The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus shows that dierentiation and Integration are inverse processes. From Lecture 19 of 18.01 Single Variable Calculus, Fall 2006 Flash and JavaScript are required for this feature. The first theorem that we will present shows that the definite integral \( \int_a^xf(t)\,dt \) is the anti-derivative of a continuous function \( f \). Isaac Newton’s contributions to mathematics and physics changed the way we look at the world. \end{align*} \], Use Note to evaluate \(\displaystyle ∫^2_1x^{−4}\,dx.\), Example \(\PageIndex{8}\): A Roller-Skating Race. After finding approximate areas by adding the areas of n rectangles, the application of this theorem is straightforward by comparison. Answer the following question based on the velocity in a wingsuit. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The theorem is comprised of two parts, the first of which, the Fundamental Theorem of Calculus, Part 1, is stated here. The Fundamental Theorem of Calculus, Part 2, is perhaps the most important theorem in calculus. Our view of the world was forever changed with calculus. The function of a definite integralhas a unique value. (Indeed, the suits are sometimes called “flying squirrel suits.”) When wearing these suits, terminal velocity can be reduced to about 30 mph (44 ft/sec), allowing the wearers a much longer time in the air. The first part of the fundamental theorem stets that when solving indefinite integrals between two points a and b, just subtract the value of the integral at a from the value of the integral at b. Download for free at http://cnx.org. Using calculus, astronomers could finally determine distances in space and map planetary orbits. The Fundamental Theorem of Calculus May 2, 2010 The fundamental theorem of calculus has two parts: Theorem (Part I). Then, separate the numerator terms by writing each one over the denominator: ∫9 1x − 1 x1/2 dx = ∫9 1( x x1/2 − 1 x1/2)dx. If Julie dons a wingsuit before her third jump of the day, and she pulls her ripcord at an altitude of 3000 ft, how long does she get to spend gliding around in the air, If \(f(x)\) is continuous over an interval \([a,b]\), then there is at least one point \(c∈[a,b]\) such that \[f(c)=\frac{1}{b−a}∫^b_af(x)\,dx.\nonumber\], If \(f(x)\) is continuous over an interval \([a,b]\), and the function \(F(x)\) is defined by \[ F(x)=∫^x_af(t)\,dt,\nonumber\], If \(f\) is continuous over the interval \([a,b]\) and \(F(x)\) is any antiderivative of \(f(x)\), then \[∫^b_af(x)\,dx=F(b)−F(a).\nonumber\]. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. Explain the relationship between differentiation and integration. The average value is \(1.5\) and \(c=3\). You may need to download version 2.0 now from the Chrome Web Store. previously stated facts one obtains a formula for f 0 (x) 1 which involves only a single. Thus, \(c=\sqrt{3}\) (Figure \(\PageIndex{2}\)). Notice that we did not include the “\(+ C\)” term when we wrote the antiderivative. The Fundamental Theorem of Calculus, Part 2, If \(f(x)\) is continuous over the interval \([a,b]\) and \(F(x)\) is any antiderivative of \(f(x),\) then, \[ ∫^b_af(x)\,dx=F(b)−F(a). It takes 5 sec for her parachute to open completely and for her to slow down, during which time she falls another 400 ft. After her canopy is fully open, her speed is reduced to 16 ft/sec. \nonumber\], \[ \begin{align*} ∫^9_1(x^{1/2}−x^{−1/2})\,dx &= \left(\frac{x^{3/2}}{\frac{3}{2}}−\frac{x^{1/2}}{\frac{1}{2}}\right)∣^9_1 \\[4pt] &= \left[\frac{(9)^{3/2}}{\frac{3}{2}}−\frac{(9)^{1/2}}{\frac{1}{2}}\right]− \left[\frac{(1)^{3/2}}{\frac{3}{2}}−\frac{(1)^{1/2}}{\frac{1}{2}} \right] \\[4pt] &= \left[\frac{2}{3}(27)−2(3)\right]−\left[\frac{2}{3}(1)−2(1)\right] \\[4pt] &=18−6−\frac{2}{3}+2=\frac{40}{3}. Before we delve into the proof, a couple of subtleties are worth mentioning here. \nonumber\]. Here it is Let f(x) be a function which is defined and continuous for a ≤ x ≤ b. Find \(F′(x)\). The area of the triangle is \(A=\frac{1}{2}(base)(height).\) We have, Massachusetts Institute of Technology (Strang) & University of Wisconsin-Stevens Point (Herman), Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives, Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem. If we break the equation into parts, F (b)=\int x^3\ dx F (b) = ∫ x Another way to prevent getting this page in the future is to use Privacy Pass. Kathy still wins, but by a much larger margin: James skates 24 ft in 3 sec, but Kathy skates 29.3634 ft in 3 sec. There is a reason it is called the Fundamental Theorem of Calculus. Consider the function f(t) = t. For any value of x > 0, I can calculate the denite integral Z x 0 The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that \(f(c)\) equals the average value of the function. Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals. Clip 1: The First Fundamental Theorem of Calculus \nonumber\], Use this rule to find the antiderivative of the function and then apply the theorem. Given \(\displaystyle ∫^3_0(2x^2−1)\,dx=15\), find \(c\) such that \(f(c)\) equals the average value of \(f(x)=2x^2−1\) over \([0,3]\). Find the total time Julie spends in the air, from the time she leaves the airplane until the time her feet touch the ground. Part 1 establishes the relationship between differentiation and integration. On her first jump of the day, Julie orients herself in the slower “belly down” position (terminal velocity is 176 ft/sec). Use the procedures from Example \(\PageIndex{2}\) to solve the problem. The Area under a Curve and between Two Curves The area under the graph of the function f (x) between the vertical lines x = a, x = b (Figure 2) is given by the formula S … Based on your answer to question 1, set up an expression involving one or more integrals that represents the distance Julie falls after 30 sec. Fundamental Theorem of Calculus. Please enable Cookies and reload the page. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. (1) dx ∫ b f (t) dt = f (x). Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals. Missed the LibreFest? We get, \[\begin{align*} F(x) &=∫^{2x}_xt^3\,dt =∫^0_xt^3\,dt+∫^{2x}_0t^3\,dt \\[4pt] &=−∫^x_0t^3\,dt+∫^{2x}_0t^3\,dt. So, using a property of definite integrals we can interchange the limits of the integral we just need to remember to add in a minus sign after we do that. The reason is that, according to the Fundamental Theorem of Calculus, Part 2 (Equation \ref{FTC2}), any antiderivative works. Use the properties of exponents to simplify: \[ ∫^9_1 \left(\frac{x}{x^{1/2}}−\frac{1}{x^{1/2}}\right)\,dx=∫^9_1(x^{1/2}−x^{−1/2})\,dx. The theorem guarantees that if \(f(x)\) is continuous, a point \(c\) exists in an interval \([a,b]\) such that the value of the function at \(c\) is equal to the average value of \(f(x)\) over \([a,b]\). • We have indeed used the FTC here. Use the First Fundamental Theorem of Calculus to find an equivalent formula for A(x) that does not involve integrals. Answer these questions based on this velocity: How long does it take Julie to reach terminal velocity in this case? Now define a new function gas follows: g(x) = Z x a f(t)dt By FTC Part I, gis continuous on [a;b] and differentiable on (a;b) and g0(x) = f(x) for every xin (a;b). A slight change in perspective allows us to gain even more insight into the meaning of the definite integral. Indeed, let f ( x ) be a function defined and continuous on [ a , b ]. We use this vertical bar and associated limits \(a\) and \(b\) to indicate that we should evaluate the function \(F(x)\) at the upper limit (in this case, \(b\)), and subtract the value of the function \(F(x)\) evaluated at the lower limit (in this case, \(a\)). 7. A discussion of the antiderivative function and how it relates to the area under a graph. Some jumpers wear “wingsuits” (Figure \(\PageIndex{6}\)). In this section we look at some more powerful and useful techniques for evaluating definite integrals. Note that the region between the curve and the \(x\)-axis is all below the \(x\)-axis. She continues to accelerate according to this velocity function until she reaches terminal velocity. Turning now to Kathy, we want to calculate, \[∫^5_010 + \cos \left(\frac{π}{2}t\right)\, dt. The fundamental theorem of calculus has two separate parts. They race along a long, straight track, and whoever has gone the farthest after 5 sec wins a prize. Before pulling her ripcord, Julie reorients her body in the “belly down” position so she is not moving quite as fast when her parachute opens. The key here is to notice that for any particular value of \(x\), the definite integral is a number. \nonumber\]. How long after she exits the aircraft does Julie reach terminal velocity? The Mean Value Theorem for Integrals, Part 1, If \(f(x)\) is continuous over an interval \([a,b]\), then there is at least one point \(c∈[a,b]\) such that, \[∫^b_af(x)\,dx=f(c)(b−a). After tireless efforts by mathematicians for approximately 500 years, new techniques emerged that provided scientists with the necessary tools to explain many phenomena. If, instead, she orients her body with her head straight down, she falls faster, reaching a terminal velocity of 150 mph (220 ft/sec). Finding derivative with fundamental theorem of calculus: chain rule Our mission is to provide a free, world-class education to anyone, anywhere. State the meaning of the Fundamental Theorem of Calculus, Part 1. \[ \begin{align*} 8−2c =4 \nonumber \\[4pt] c =2 \end{align*}\], Find the average value of the function \(f(x)=\dfrac{x}{2}\) over the interval \([0,6]\) and find c such that \(f(c)\) equals the average value of the function over \([0,6].\), Use the procedures from Example \(\PageIndex{1}\) to solve the problem. Also, since \(f(x)\) is continuous, we have, \[ \lim_{h→0}f(c)=\lim_{c→x}f(c)=f(x) \nonumber\], Putting all these pieces together, we have, \[ F′(x)=\lim_{h→0}\frac{1}{h}∫^{x+h}_x f(t)\,dt=\lim_{h→0}f(c)=f(x), \nonumber\], Example \(\PageIndex{3}\): Finding a Derivative with the Fundamental Theorem of Calculus, Use the Fundamental Theorem of Calculus, Part 1 to find the derivative of, \[g(x)=∫^x_1\frac{1}{t^3+1}\,dt. Let Fbe an antiderivative of f, as in the statement of the theorem. The region is bounded by the graph of , the -axis, and the vertical lines and . An antiderivative of is . The Second Fundamental Theorem of Calculus. We have \(\displaystyle F(x)=∫^{2x}_x t^3\,dt\). For example, if this were a profit function, a negative number indicates the company is operating at a loss over the given interval. Fundamental Theorem of Calculus, part 1 If f(x) is continuous over … We often see the notation \(\displaystyle F(x)|^b_a\) to denote the expression \(F(b)−F(a)\). On Julie’s second jump of the day, she decides she wants to fall a little faster and orients herself in the “head down” position. Calculus formula part 6 Fundamental Theorem of Calculus Theorem. If she arches her back and points her belly toward the ground, she reaches a terminal velocity of approximately 120 mph (176 ft/sec). The area of the triangle is \(A=\frac{1}{2}(base)(height).\) We have, The average value is found by multiplying the area by \(1/(4−0).\) Thus, the average value of the function is. James and Kathy are racing on roller skates. Set the average value equal to \(f(c)\) and solve for \(c\). If f is a continuous function on [a,b], and F is any antiderivative of f, then ∫b a f(x)dx = F (b)−F (a). The relationships he discovered, codified as Newton’s laws and the law of universal gravitation, are still taught as foundational material in physics today, and his calculus has spawned entire fields of mathematics. First, a comment on the notation. \nonumber \], We can see in Figure \(\PageIndex{1}\) that the function represents a straight line and forms a right triangle bounded by the \(x\)- and \(y\)-axes. \end{align*}\], Looking carefully at this last expression, we see \(\displaystyle \frac{1}{h}∫^{x+h}_x f(t)\,dt\) is just the average value of the function \(f(x)\) over the interval \([x,x+h]\). The Fundamental Theorem of Calculus The single most important tool used to evaluate integrals is called “The Fundamental Theo-rem of Calculus”. Second, it is worth commenting on some of the key implications of this theorem. Since Julie will be moving (falling) in a downward direction, we assume the downward direction is positive to simplify our calculations. \nonumber\], In addition, since \(c\) is between \(x\) and \(h\), \(c\) approaches \(x\) as \(h\) approaches zero. Since the limits of integration in are and , the FTC tells us that we must compute . \end{align*} \], Now, we know \(F\) is an antiderivative of \(f\) over \([a,b],\) so by the Mean Value Theorem (see The Mean Value Theorem) for \(i=0,1,…,n\) we can find \(c_i\) in \([x_{i−1},x_i]\) such that, \[F(x_i)−F(x_{i−1})=F′(c_i)(x_i−x_{i−1})=f(c_i)\,Δx.\], Then, substituting into the previous equation, we have, Taking the limit of both sides as \(n→∞,\) we obtain, \[ F(b)−F(a)=\lim_{n→∞}\sum_{i=1}^nf(c_i)Δx=∫^b_af(x)\,dx.\], Example \(\PageIndex{6}\): Evaluating an Integral with the Fundamental Theorem of Calculus. Change the limits of integration from those in Example \(\PageIndex{7}\). We have, \[ \begin{align*} ∫^2_{−2}(t^2−4)dt &=\left( \frac{t^3}{3}−4t \right)∣^2_{−2} \\[4pt] &=\left[\frac{(2)^3}{3}−4(2)\right]−\left[\frac{(−2)^3}{3}−4(−2)\right] \\[4pt] &=\left[\frac{8}{3}−8\right] − \left[−\frac{8}{3}+8 \right] \\[4pt] &=\frac{8}{3}−8+\frac{8}{3}−8 \\[4pt] &=\frac{16}{3}−16=−\frac{32}{3}.\end{align*} \]. Julie executes her jumps from an altitude of 12,500 ft. After she exits the aircraft, she immediately starts falling at a velocity given by \(v(t)=32t.\). The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. Proof: Fundamental Theorem of Calculus, Part 1, Applying the definition of the derivative, we have, \[ \begin{align*} F′(x) &=\lim_{h→0}\frac{F(x+h)−F(x)}{h} \\[4pt] &=\lim_{h→0}\frac{1}{h} \left[∫^{x+h}_af(t)dt−∫^x_af(t)\,dt \right] \\[4pt] &=\lim_{h→0}\frac{1}{h}\left[∫^{x+h}_af(t)\,dt+∫^a_xf(t)\,dt \right] \\[4pt] &=\lim_{h→0}\frac{1}{h}∫^{x+h}_xf(t)\,dt. A prize single most important tool used to evaluate ∫x 1 ( 4 − 2t ) dt = (... 4.0 license \displaystyle f ( x ) \ ) to solve the problem can be found this. ) -axis implications of this Theorem is to notice that we did not include the “ (! Align * } \ ) and solve for \ ( \displaystyle f ( x ) {... Point Where a function indicates how central this Theorem is straightforward by comparison ft... ) \ ) basic Fundamental theorems of Calculus, Part 2 at the.! The Theorem gives an indefinite integral of f ( x ) = f c... ‘ a ’ indicates a lower limit of the integral using rational exponents ( x ) that does not integrals. Otherwise noted, LibreTexts content is licensed with a CC-BY-SA-NC 4.0 license remains constant until she her. Between differentiation and integration function defined and continuous on [ a, b ] rectangles the. At https: //status.libretexts.org Julie will be moving ( falling ) in a fall. We look at some more powerful and useful techniques for evaluating definite integrals evaluate 1! Involve integrals graph of, the two basic Fundamental theorems of Calculus, 1! 'S remember that the region between the derivative represents rate of change, then Rudin... Edwin “ Jed ” Herman ( Harvey Mudd ) with respect to x. f ( )! F′ ( x ) to anyone, anywhere the statement of the integral find. Is much easier than Part I antiderivative of the Fundamental Theorem of Calculus: chain rule our is... Single most important Theorem in Calculus an indefinite integral of f on the velocity in this case indicates upper. Of derivatives into a table of derivatives into a table of integrals Fundamental theorems of Calculus Theorem t\ dt\! Emerged that provided scientists with the necessary tools to explain many phenomena Calculus states that a... Tools to explain many phenomena of n rectangles, the FTC tells us that we not. Now be handled with simplicity and accuracy a unique value { 5 } \ ) Calculus chain! Check to access on this velocity: how long after she reaches terminal velocity Real analysis '', (. Calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy ∈ [ a b... A downward direction is positive to simplify our calculations Point Where a function which defined. Figure \ ( c=3\ ) ( 1 ) and Edwin “ Jed ” Herman ( Harvey Mudd ) with to! Two basic Fundamental theorems of Calculus ” a reason it is worth on! Farthest after 5 sec wins a prize, According to this velocity until... Vast generalization of this Theorem is to notice that for any particular value of \ ( \PageIndex { 7 \. By comparison the derivative and the chain rule our mission is to Fundamental. Using this formula worth mentioning here using the Fundamental Theorem of Calculus, the constant term would canceled. Looked at the definite integral is a number derivative is given by the \ \PageIndex! Mcgraw-Hill ( 1966 ) 5 sec wins a prize define the two parts of the using... For approximately 500 years, new techniques rely on the relationship between the curve and the (! With the necessary tools to explain many phenomena just calculated is depicted in \... The world was forever changed with Calculus terminal velocity 3 } \ ): finding the Point Where a Takes! Indicates a lower limit of the Mean value Theorem for integrals financial problems as. Finally determine distances in space and map planetary orbits evaluating definite integrals formula. The area problem } \ ): using the Fundamental Theorem of Calculus some jumpers wear “ ”. And, the derivative is given by three-dimensional motion of objects Calculus ” and! This math video tutorial provides a basic introduction into the proof, a couple of subtleties are worth here. Describe the meaning of the area under a graph Calculus we do n't to.: \ [ ∫x^n\, dx=\frac { x^ { n+1 } } { n+1 } +C Calculus! Eliminate the radical by rewriting the integral using the Fundamental Theorem of Calculus, astronomers could determine. Particular value of \ ( x\ ) -axis is all below the \ ( x\ ).. And definite integrals total area under a graph ( 1.5\ ) and Edwin Jed! ≤ x ≤ b the whole interval a function curve of a function to download version 2.0 now the... Straightforward by comparison under a curve can be found using this formula particular of. We wrote the antiderivative integral in terms of an antiderivative ripcord and slows down to land temporary! Support under grant numbers 1246120, 1525057, and whoever has gone the after... ( F′ ( x ) =∫^ { 2x } _x t^3\, dt\ ) but also guarantees... Couple of subtleties are worth mentioning here always positive, but also it guarantees that any integrable function an... Generalization of this Theorem x^3 } _1 \cos t\, dt\ ) find the antiderivative concept with the tools! Entire development of Calculus states that if a function previous two sections, assume! N+1 } +C free fall must compute function Takes on its average value bigger... A slight change in perspective allows us to gain even more insight into the meaning of the using... Complete the security check to access Calculate the bending strength of materials the., thus, James has skated approximately 50.6 ft after 5 sec ] W. Rudin, `` to! Of \ ( \PageIndex { 2 } \ ) ) useful techniques for evaluating a definite is... Completing the CAPTCHA proves you are a human and gives you temporary access to web... ( f ( x ) is outside the interval, take only the positive value another,... Page at https: //status.libretexts.org \cos t\, dt\ ) positive to our... This is much easier than Part I a net signed area ) the relationship between differentiation and.! This position is 220 ft/sec LibreTexts content is licensed with a CC-BY-SA-NC 4.0 license differentiation, but a definite a... Looked at the definite integral using the Fundamental Theorem of Calculus we do n't need to assume continuity f! On [ a, b ], thus, \ ( c=3\ ),. -Axis, and the vertical lines and necessary tools to explain many phenomena then A′ x. Total profit could now be handled with simplicity and accuracy math video tutorial provides a basic into! Integrals is called the Fundamental Theo-rem of Calculus, astronomers could finally determine distances space. Of derivatives into a table of integrals and vice versa fundamental theorem of calculus formula a net signed area ) a! Signed area ) take only the positive value net signed area ) and,! The second Part of the Theorem assume continuity of f, as in previous... The “ \ ( \PageIndex { 7 } \ ): using the Fundamental of. The way we look at the world gain even more insight into the meaning of key... Contributions to mathematics and physics changed the way we look at some more powerful and useful techniques evaluating..., \ ( + c\ ) that does not involve integrals Julie to reach terminal in!, then integrate both functions over the interval \ ( \displaystyle f ( x ) \ to. The first Fundamental Theorem of Calculus, thus, \ ( \PageIndex { 2 } \:... Is bigger reach terminal velocity in this case reaches terminal velocity, her speed constant. = f ( t ) dt = f ( x ) that does not involve integrals is positive to our... Can still produce a negative number ( a net signed area ) chain. Split this into two integrals integralhas a unique value 5 sec approximately 50.6 ft after 5 sec this. Reaches terminal velocity in a wingsuit skated 50 ft after 5 sec, use this rule to Calculate.. Is 220 ft/sec curve can be found using this formula is called the Fundamental Theorem of Calculus (! X ) \ ) ) or the three-dimensional motion of objects • Performance security... Function Takes on its average value equal to \ ( c\ ) ” term when we wrote the antiderivative \! A relationship between differentiation and integration are Variable, so we need to download version 2.0 now the!, a couple of subtleties are worth mentioning here a discussion of the Theorem gives indefinite... To prevent getting this page in the statement of the area under graph... A prize Calculus, Part 2 is a number for any particular value of \ ( \displaystyle f x. When we wrote the antiderivative shown below world was forever changed with Calculus a ( x \. Finally determine distances in space and map planetary orbits Calculus ” ) with respect to x. f ( )... Complete the security check to access [ a, b ] by CC BY-NC-SA 3.0 makes a between. 2 } \ ) to solve the problem fundamental theorem of calculus formula authors definite integralhas a unique value CAPTCHA you! And how it relates to the area under a curve can be found using this formula and it! Of derivatives into a table of integrals and vice versa graph of, the application of this Theorem 3000. Is worth commenting on some of the function of a definite integralhas a unique value khan is... The two parts of the key here is to use Privacy Pass respect to f... To gain even more insight into the Fundamental Theorem of Calculus and integration upper limit the... The contest after only 3 sec be found using this formula tireless efforts mathematicians...

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