Question 4 (10 points) Evaluate $$ \int \sec ^{2} x an ^{5} x d x $$

Question

Question 4 (10 points) Evaluate $$ \int \sec ^{2} x 	an ^{5} x d x $$

UpStudy AI · Accepted Answer

Let $$ u = \tan x $$. Then, the derivative is

$$
\frac{du}{dx} = \sec^2 x \quad \text{or} \quad du = \sec^2 x \, dx.
$$

Thus, the integral transforms as

$$
\int \sec^2 x \tan^5 x \, dx = \int u^5 \, du.
$$

Now, integrating $$ u^5 $$ with respect to $$ u $$:

$$
\int u^5 \, du = \frac{u^6}{6} + C.
$$

Substituting back $$ u = \tan x $$ gives

$$
\frac{\tan^6 x}{6} + C.
$$

Thus, the evaluated integral is

$$
\boxed{\frac{\tan^6 x}{6} + C}.
$$
The integral of $$ \sec^2 x \tan^5 x \, dx $$ is $$ \frac{\tan^6 x}{6} + C $$.

Question 4 (10 points) Evaluate \( \int \sec ^{2} x \tan ^{5} x d x \)