6.9 Integrating Using Substitution

Learning objectives

—FUN-6.D—For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate…• FUN-6.D For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals.• FUN-6.D.1 Substitution of variables is a technique for finding antiderivatives.• FUN-6.D.2 For a definite integral, substitution of variables requires corresponding changes to the limits of integration.• Enduring understanding FUN-6: Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Use substitution when an integrand contains a composite expression together with its derivative, possibly differing by a constant factor. Setting the inner expression equal to $u$ reverses the chain rule and turns the integral into a familiar function of one variable.

Choose $u$ as the inner expression; compute $du$ ; rewrite every factor and the differential in terms of $u$ ; integrate; then check by differentiating. For an indefinite integral, substitute the original expression back and add $C$ . For a definite integral, convert both bounds using the same substitution before evaluating.

For $\int 6x(3x^2+1)^4\,dx$ , let $u=3x^2+1$ , so $du=6x\,dx$ . Then $\int u^4\,du=u^5/5+C$ , giving $(3x^2+1)^5/5+C$ . Differentiation returns the original integrand.

For $\int_0^1 2x e^{x^2}\,dx$ , let $u=x^2$ and $du=2x\,dx$ . The original bounds become $u(0)=0$ and $u(1)=1$ , so the integral is $\int_0^1 e^u\,du=[e^u]_0^1=e-1$ .

Do not leave both $x$ and $u$ in the transformed integrand. For a definite integral, either change the bounds and finish entirely in $u$ , or find an antiderivative, return to $x$ , and use the original bounds. Mixing new bounds with an $x$ -expression gives an invalid evaluation.

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