Evaluation of Definite Integrals by Substitution

Evaluation of Definite Integrals by Substitution

The method of substitution, used extensively for indefinite integrals, adapts to evaluate definite integrals directly. The key insight: when you change the variable of integration, you must also change the limits of integration to match the new variable. This eliminates the need to substitute back to the original variable before applying the limits.

Two Approaches

Approach 1 (resubstitute):

1. Consider the indefinite integral $\int f(x)\,dx$ and make a substitution to reduce it to a known form.
2. Integrate with respect to the new variable, omitting the constant of integration.
3. Resubstitute to express the result in terms of the original variable.
4. Evaluate at $x = a$ and $x = b$ and take the difference.

Approach 2 (change the limits — direct method):

1. Choose the substitution $t = g(x)$ that simplifies the integrand, and express $dx$ in terms of $dt$ by differentiating it.
2. Change the limits: if $x = a$ then $t = g(a)$; if $x = b$ then $t = g(b)$.
3. Rewrite the entire integral in terms of $t$ with the new limits, and evaluate $\int_{g(a)}^{g(b)} g(t)\,dt$ directly.

Exercise 7.9 — Key Substitutions

1. $\int_0^1 \frac{x}{1+x^2}\,dx$ — Put $t = 1+x^2$
2. $\int_0^{\pi/2} \sin^5\phi\cos\phi\,d\phi$ — Put $t = \sin\phi$
3. $\int_0^1 \frac{2\sin^{-1}x}{1+x^2}\,dx$ — Put $t = \sin^{-1}x$
4. $\int_0^2 x\sqrt{x+2}\,dx$ — Put $x+2 = t^2$
5. $\int_0^{\pi/2} \frac{\sin x}{1+\cos^2 x}\,dx$ — Put $t = \cos x$
6. $\int_0^2 \frac{dx}{x+4-x^2}$ — Complete the square
7. $\int_{-1}^1 \frac{dx}{x^2+2x+5}$ — Complete the square
8. $\int_1^2 \left(\frac{1}{x} - \frac{1}{2x^2}\right)e^{2x}\,dx$ — Put $t = 2x$

Common Pitfalls to Avoid

• Forgetting to change limits: in the direct method, always compute the new limits before integrating. …