L> The unknown Integral
The indefinite IntegralIntegrationWe begin with a question.Question: list two features F(x) such the F"(x) = xAnswer: 1/2 x2and 1/2 x2 + 3We can see the if F"(x) = x climate F(x) = 1/2 x2 + C because that some constant C.We contact F(x) the antiderivativeor integral that f(x) and also write
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In general, ifF"(x) = f(x) then we create
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From the derivative formuladxn = nxn-1 dxWe get theintegral formula

The power Rule

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Justlike through derivatives, to find an antiderivative that a amount or difference, us cantake the antiderivative of each term. Likewise like derivatives, theantiderivative the the product or quotient is not easily found.

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ExampleWhichof these has simple to discover antiderivativeA.8x3 - 6xB.xx5 + 2SolutionWe can uncover the antiderivative of component A. Easily, by finding theantiderivative that 8x3 and 6xseparately. The antiderivative is8(1/4 x4) - 6(1/2 x2) + C = 2x4 -3x2 + CB. This one, on the other hand,is aquotient. We carry out not have actually a means of finding itsantiderivative.ExerciseFind the following integrals: (x + x2 )dx 1/ x2 dx (12 x )2 dx 1 /
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dx (1 - 2x)20 dx specific SolutionsWe have seen the an integral produces a entirety family of remedies parameterizedby C. In many applications, us are offered an early stage or other conditionand hence find the value of C. The antiderivative with well-known C is calleda particular solution.ExampleFind a equipment to F"(x)= 4x - 3given that F(1) =2Solution: We an initial find an antiderivative: F(x) =2x2 - 3x + CNow plug in 1 because that x and also 2 for F to get: 2= 2(1)2 - 3(1) + C = -1 + C So that C = 3. The certain solution is F(x) = 2x2 - 3x + 3.Example
Find the equipment to the differential equation dy/dx = 3x2- 4x + 2SolutionWe discover the antiderivative of 3x2- 4x + 2We can discover this antiderivative by recognize the antiderivative of x2,x, and also 2 separately. 3(1/3 x2)- 4(1/2 x2) + 2x = x2- 2x2 + 2xNotice that due to the fact that the derivative the a continuous is zero, adding a consistent of anantiderivative outcomes in one more antiderivative because that the exact same function. Wecan compose the final answer as x2 -2x2 + 2x + Cwhere C represents any type of constant.ApplicationsSince the acceleration of heaviness is a consistent a = 32, we have the right to derivethe physics equations.ExampleSuppose that us kicka football through an initial upward velocity of100feet per 2nd how long will it take to hit the ground?SolutionWe have actually v(t) =-32 dt = -32t + C v(0) = 100 = C s(t) =(-32t + 100)dt =-16t2 + 100t + C s(0) =0 = Chence s(t) =-16t2 + 100t = t(-16t + 100)So the s(t) =0 as soon as -16t + 100 = 0 or t =100/16 = 6.25It will certainly take 6.25 seconds to fight the ground.ExampleSuppose the marginal revenue for a ski resort is M = 50 -0.01 xAnd expect that in ~ $50 every ticket, the ski resortwill have actually 2,000 skiers.Find the need equation.SolutionSince the marginal revenue is the derivative the the revenue, the revenue is theantiderivative the the marginal revenue.

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R =(50 -0.01x)dx= 50x - 0.005 x2 + CThe revenue is equal tothe price times the quantity. The is50x - 0.005 x2 + C = pxNowfind C by noting that as soon as p= 50, x = 2,000.50(2,000) - 0.0005 (2,000)2 + C = (50)(2,000)80,000 + C = 1,000,000C = 920,000Substituting the Cinto our equation and also dividing by x offers the demand equationp = 50 - 0.005 x + 920,000/xBackto the mathematics 116 residence PageBack come the MathDepartment Home