Questions: Evaluate the indefinite integral ∫ 8 tan(13x) sec^6(13x) dx. Use C for the constant of integration. Write the exact answer, Do not round.

Transcript text: Evaluate the indefinite integral $\int 8 \tan (13 x) \sec ^{6}(13 x) d x$. Use $C$ for the constant of integration. Write the exact answer, Do not round.

Solution

Solution Steps

To evaluate the indefinite integral $\int 8 \tan (13 x) \sec ^{6}(13 x) d x$, we can use a substitution method. Let $u = 13x$, then $du = 13dx$. This substitution will simplify the integral, making it easier to solve.

Step 1: Substitution

To evaluate the integral $\int 8 \tan (13 x) \sec ^{6}(13 x) \, dx$, we start by using the substitution $u = 13x$. Then, $du = 13 \, dx$ or $dx = \frac{du}{13}$.

Step 2: Simplify the Integral

Substituting $u$ and $dx$ into the integral, we get: \[ \int 8 \tan(u) \sec^6(u) \cdot \frac{du}{13} \] This simplifies to: \[ \frac{8}{13} \int \tan(u) \sec^6(u) \, du \]

Step 3: Integrate

We recognize that the integral of $\tan(u) \sec^6(u)$ can be solved directly. The integral of $\tan(u) \sec^6(u)$ is $\frac{1}{2} \sec^4(u)$. Therefore: \[ \frac{8}{13} \int \tan(u) \sec^6(u) \, du = \frac{8}{13} \cdot \frac{1}{2} \sec^4(u) = \frac{4}{13} \sec^4(u) \]

Step 4: Substitute Back

Substitute $u = 13x$ back into the expression: \[ \frac{4}{13} \sec^4(13x) \]

Step 5: Add the Constant of Integration

Finally, we add the constant of integration $C$: \[ \frac{4}{13} \sec^4(13x) + C \]