Question 9.2.1.
For each of the indefinite integrals below, decide whether the integral can be evaluated: immediately because it's a basic integral, using \(u\)-substitution, integration by parts, with multiple techniques, or if none of those will work.
\(\displaystyle \int \frac{1}{1+x^2} \, dx\)
- Basic Integral
- u-Substitution
- Integration by Parts
- A combination of techniques
- None of the above
\(\displaystyle \int \frac{x}{1+x^2} \, dx\)
- Basic Integral
- u-Substitution
- Integration by Parts
- A combination of techniques
- None of the above
\(\displaystyle \int \frac{2x+3}{1+x^2} \, dx\)
- Basic Integral
- u-Substitution
- Integration by Parts
- A combination of techniques
- None of the above
\(\displaystyle \int \frac{e^x}{1+(e^x)^2} \, dx\)
- Basic Integral
- u-Substitution
- Integration by Parts
- A combination of techniques
- None of the above
\(\displaystyle \int \frac{1}{x^2-1} \, dx\)
- Basic Integral
- u-Substitution
- Integration by Parts
- A combination of techniques
- None of the above
Hint.
Note that \(\displaystyle \frac{2x+3}{1+x^2}\) can be rewritten as \(\displaystyle \frac{2x}{1+x^2} +\frac{3}{1+x^2}\text{.}\)