Focuses on the step-by-step mechanical process of solving indefinite integrals using u-substitution. Includes examples with polynomial, radical, and basic transcendental functions, and handling constant multipliers.
Consider \(\int x \sqrt{x^2+9} \, dx\). Let \(u = x^2+9\). What is \(du\)?
\(du = 2x \, dx\).
For \(\int x \sqrt{x^2+9} \, dx\), we have \(u = x^2+9\) and \(du = 2x \, dx\). The term \(x \, dx\) is in the integral, but not \(2x \, dx\). How do you handle this?
Solve for \(x \, dx\) in the differential: \(\frac{1}{2} du = x \, dx\). Then, substitute \(\frac{1}{2} du\) for \(x \, dx\) in the integral.
Rewrite \(\int x \sqrt{x^2+9} \, dx\) in terms of \(u\), given \(u=x^2+9\).
Substituting \(\sqrt{u}\) for \(\sqrt{x^2+9}\) and \(\frac{1}{2}du\) for \(x\,dx\) gives: $$\int \sqrt{u} \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^{1/2} \, du$$
Find the final solution for \(\int x \sqrt{x^2+9} \, dx\).
What is a good choice for \(u\) in the integral \(\int \frac{5}{(3x-8)^2} \, dx\)?
Let \(u = 3x - 8\). The derivative is \(du = 3 \, dx\).
Solve \(\int \frac{5}{(3x-8)^2} \, dx\) using u-substitution.
Let \(u = 3x-8\), so \(du = 3\,dx\) and \(dx = \frac{1}{3}du\). The integral becomes \(\int \frac{5}{u^2} \frac{1}{3}du = \frac{5}{3} \int u^{-2}du = \frac{5}{3}(-u^{-1}) + C = -\frac{5}{3u} + C\). Back-substituting gives \(-\frac{5}{3(3x-8)} + C\).
For the integral \(\int \sin(x)\cos(x) \, dx\), what are two possible choices for \(u\)?
You can choose either \(u = \sin(x)\) (with \(du = \cos(x) \, dx\)) or \(u = \cos(x)\) (with \(du = -\sin(x) \, dx\)). Both will lead to a correct answer.
