(1 point) Evaluate the following indefinite integral using the substitution $u=8 x+13$.
\[
\int \frac{10}{(8 x+13)^{6}} d x=
\]
Answer
\(\boxed{-\frac{1}{4} (8x + 13)^{-5} + C}\) is the solution to the integral, where \(C\) is the constant of integration.
Step 1 ：Identify the substitution: \(u = 8x + 13\).
Step 2 ：Differentiate the substitution: \(du = 8dx\).
Step 3 ：Solve for \(dx\): \(dx = du/8\).
Step 4 ：Substitute \(u\) and \(dx\) into the integral: \(\int \frac{10}{u^{6}} \cdot \frac{du}{8} = \frac{10}{8} \int \frac{1}{u^{6}} du\).
Step 5 ：Simplify the integral: \(\frac{5}{4} \int u^{-6} du\).
Step 6 ：Evaluate the integral: \(\frac{5}{4} \cdot \left(-\frac{u^{-5}}{5}\right) = -\frac{1}{4} u^{-5}\).
Step 7 ：Substitute \(u = 8x + 13\) back into the result: \(-\frac{1}{4} (8x + 13)^{-5}\).
Step 8 ：Check the result by differentiating \(-\frac{1}{4} (8x + 13)^{-5}\) with respect to \(x\) and confirming that it gives back the original integrand \(\frac{10}{(8x + 13)^6}\).
Step 9 ：\(\boxed{-\frac{1}{4} (8x + 13)^{-5} + C}\) is the solution to the integral, where \(C\) is the constant of integration.
