Question

find the indefinite integral.
int(3 x - 5)^{-6} d x
int(3 x - 5)^{-6} d x =

Explanation:

Step1: Use substitution method

Let \(u = 3x - 5\), then \(du=3dx\), and \(dx=\frac{1}{3}du\).

Step2: Substitute into the integral

The integral \(\int(3x - 5)^{-6}dx\) becomes \(\int u^{-6}\cdot\frac{1}{3}du\).

Step3: Integrate with respect to \(u\)

Using the power - rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(\int\frac{1}{3}u^{-6}du\), we have \(\frac{1}{3}\cdot\frac{u^{-6 + 1}}{-6+1}+C=\frac{1}{3}\cdot\frac{u^{-5}}{-5}+C=-\frac{1}{15}u^{-5}+C\).

Step4: Substitute back \(u = 3x - 5\)

We get \(-\frac{1}{15}(3x - 5)^{-5}+C\).

Answer:

\(-\frac{1}{15}(3x - 5)^{-5}+C\)