Flashcards in 9.2.2 Integrating Polynomials by Substitution Deck (12):
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.
- 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
- 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.
(x−1)^4/3 / 4/3 + C
Solve the following integral:∫2x^2(x^3+3)^3/2dx.
2(7+x^5)^3/2 / 15+C
(x^4−1)^6 / 24+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