Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \tan ^{7} \frac{x}{2} \sec ^{2} \frac{x}{2} d x$$

Short answer

The evaluated integral is \(\frac{\tan^8(\frac{x}{2})}{4}\).

Step-by-step solution

Perform the Substitution

Take \(u = \tan(\frac{x}{2})\), thus \( du = \sec^2(\frac{x}{2})\frac{1}{2}dx \). You can now substitute \(\tan^{7}(\frac{x}{2})\) with \(u^{7}\) and \(\sec^2(\frac{x}{2})dx \) with \(2du\).

Simplify and Evaluate Integral

The original integral simplifies to \(\int 2u^7 du\) with the substitution. Now, perform the integration, which gives \( \frac{2u^8}{8} = \frac{u^8}{4}\).

Substitute Back and Simplify

Finally, substitute \(u=\tan(\frac{x}{2})\) back into \(\frac{u^8}{4}\), resulting in \(\frac{\tan^8(\frac{x}{2})}{4}\).