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AP Calculus AB: 9.2.2 Integrating Polynomials by Substitution

Mathematics12 CardsCreated 3 months ago

This content explains how to integrate polynomial expressions using integration by substitution, especially when dealing with composite functions. It outlines the step-by-step substitution process, emphasizes correct variable handling, and highlights the importance of rewriting the final answer in terms of the original variable along with including the constant of integration.

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Integrating Polynomials by Substitution

  • The differential of an integral identifies the variable of integration.

  • Integration by substitution is a technique for finding the antiderivative of a composite function. To integrate by substitution, select an expression for u. Next, rewrite the integral in terms of u. Then simplify the integral and evaluate.

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Key Terms

Term
Definition

Integrating Polynomials by Substitution

  • The differential of an integral identifies the variable of integration.

  • Integration by substitution is a technique for findi...

note

  • When using Leibniz notation, the expression underneath the bar indicates the variable with respect to which the derivative is taken.

Integrate.∫^3√x−1dx

(x−1)^4/3 / 4/3 + C

Solve the following integral:∫2x^2(x^3+3)^3/2dx.

4/15(x^3+3)^5/2+C

Evaluate.∫x^4√7+x^5dx

2(7+x^5)^3/2 / 15+C

∫x^3(x^4−1)^5dx.

(x^4−1)^6 / 24+C

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AP Calculus AB: 9.2.2 Integrating Polynomials by Substitution
TermDefinition

Integrating Polynomials by Substitution

  • The differential of an integral identifies the variable of integration.

  • Integration by substitution is a technique for finding the antiderivative of a composite function. To integrate by substitution, select an expression for u. Next, rewrite the integral in terms of u. Then simplify the integral and evaluate.

note

  • When using Leibniz notation, the expression underneath the bar indicates the variable with respect to which the derivative is taken.

  • The same expression appears when working with integrals. Integrate with respect to the variable indicated by this expression.

  • Notice that the integrand is the product of a composite function and the derivative of its inside. The presence of a composite function is a sign that you should try integration by substitution.

  • As with derivatives, the integral of a product of two functions is not equal to the product of the integrals. Find a way to transform the integral into something you can evaluate.

  • Here, let u be the inside of the composite function.

  • Notice that the derivative of u is contained within the
    integrand.

  • Substitute so that you remove all of the x-terms from the integrand. The resulting integral is one you can evaluate with the power rule. This is how you integrate by substituting.

  • Do not forget the constant of integration.

  • When using integration by substitution, always express the answer in terms of the original variable.

Integrate.∫^3√x−1dx

(x−1)^4/3 / 4/3 + C

Solve the following integral:∫2x^2(x^3+3)^3/2dx.

4/15(x^3+3)^5/2+C

Evaluate.∫x^4√7+x^5dx

2(7+x^5)^3/2 / 15+C

∫x^3(x^4−1)^5dx.

(x^4−1)^6 / 24+C

Evaluate:∫(2x+2)(x^2+2x+1)^3dx.

(x^2+2x+1)^4/4+C

Evaluate:∫√1−xdx.

-(1−x)^3/2/ 3/2+C

Integrate.∫9dp(3p−1)^3

−3/2(3p−1)−2+C

Solve the integral.∫2y(3y^2−5)^1.7dy

(3y^2−5)^2.7 / 8.1+C

Evaluate the integral:∫x^6√(x^2+1)^5dx

3/11(x^2+1)^11/6 +C

Evaluate the integral.∫2(x+1)^5dx

−1/2(x+1)^4+C