Preview Activity 5.5.1.
For each of the indefinite integrals below, the main question is to decide whether the integral can be evaluated using \(u\)-substitution, integration by parts, a combination of the two, or neither. For integrals for which your answer is affirmative, state the substitution(s) you would use. It is not necessary to actually evaluate any of the integrals completely, unless the integral can be evaluated immediately using a familiar basic antiderivative.
(a)
\(\int x^2 \sin(x^3) \, dx\text{,}\) \(\int x^2 \sin(x) \, dx\text{,}\) \(\int \sin(x^3) \, dx\text{,}\) \(\int x^5 \sin(x^3) \, dx\)
(b)
\(\int \frac{1}{1+x^2} \, dx\text{,}\) \(\int \frac{x}{1+x^2} \, dx\text{,}\) \(\int \frac{2x+3}{1+x^2} \, dx\text{,}\) \(\int \frac{e^x}{1+(e^x)^2} \, dx\text{,}\)
(c)
\(\int x \ln(x) \, dx\text{,}\) \(\int \frac{\ln(x)}{x} \, dx\text{,}\) \(\int \ln(1+x^2) \, dx\text{,}\) \(\int x\ln(1+x^2) \, dx\text{,}\)
(d)
\(\int x \sqrt{1-x^2} \, dx\text{,}\) \(\int \frac{1}{\sqrt{1-x^2}} \, dx\text{,}\) \(\int \frac{x}{\sqrt{1-x^2}}\, dx\text{,}\) \(\int \frac{1}{x\sqrt{1-x^2}} \, dx\text{,}\)
