Sean Fitzpatrick |
---|
University of Lethbridge |
![]() |
Evaluate the following integrals:
Think about (but don't actually try to solve — yet) the integral
Sometimes a trigonometric or hyperbolic substitution can help with integration. Possible options include:
For each expression below, indicate which of the above options will work.
Case I: \(n\geq 2\) is even.
Use \(u=\tan(x), du = \sec^2(x)\,dx\text{,}\) and the identity \(1+\tan^2(x)=\sec^2(x)\text{.}\)
Examples:
Case II: \(m\geq 1\) is odd, and \(n\geq 1\)
Use \(u=\sec(x)\text{,}\) so \(du = \sec(x)\tan(x)\,dx\text{,}\) and the identity \(1+\tan^2(x)=\sec^2(x)\text{.}\)
Examples:
Case III: \(m\) is even and \(n\) is odd — the danger zone. Typically this requires multiple integrations by parts or reduction formulas.
Examples:
Examples:
\(\di\int \sqrt{9-x^2}\,dx\)
\(\di \int_{-3}^3\sqrt{9-x^2}\,dx\)
\(\di \int\frac{\sqrt{4-x^2}}{x^2}\,dx\)
Examples:
\(\di \int\sqrt{4+x^2}\,dx\)
\(\di\int\frac{x^2}{\sqrt{x^2+3}}\,dx\)
\(\di \int\frac{x}{(x^2+9)^{3/2}}\,dx\)
Examples:
\(\di \int \sqrt{x^2-16}\,dx\)
\(\di \int_4^8 \sqrt{x^2-16}\,dx\)
\(\di \int \frac{3x^2}{\sqrt{x^2-4}}\,dx\)