9.2.2 Integrating Polynomials by Substitution

Question 1

Integrating Polynomials by Substitution

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Integrate.∫^3√x−1dx

Answer 3

(x−1)^4/3 / 4/3 + C

Question 4

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

Answer 4

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

Question 5

Evaluate.∫x^4√7+x^5dx

Answer 5

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

Question 6

∫x^3(x^4−1)^5dx.

Answer 6

(x^4−1)^6 / 24+C

Question 7

Evaluate:∫(2x+2)(x^2+2x+1)^3dx.

Answer 7

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

Question 8

Evaluate:∫√1−xdx.

Answer 8

-(1−x)^3/2/ 3/2+C

Question 9

Integrate.∫9dp(3p−1)^3

Answer 9

−3/2(3p−1)−2+C

Question 10

Solve the integral.∫2y(3y^2−5)^1.7dy

Answer 10

(3y^2−5)^2.7 / 8.1+C

Question 11

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

Answer 11

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

Question 12

Evaluate the integral.∫2(x+1)^5dx

Answer 12

−1/2(x+1)^4+C