Question

find the following indefinite integral. use c for the constant of integration. \\(\int (4x^{-5} + 8) \\, dx\\)

Explanation:

Step1: Integrate term by term

We use the power rule for integration, \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)) and \(\int kdx=kx + C\) (where \(k\) is a constant).
For the first term \(4x^{-5}\):
\(\int4x^{-5}dx=4\times\frac{x^{-5 + 1}}{-5+1}\)
For the second term \(8\):
\(\int8dx = 8x\)

Step2: Simplify the first integral

Simplify \(4\times\frac{x^{-4}}{-4}\):
\(4\times\frac{x^{-4}}{-4}=-x^{-4}\)

Step3: Combine the results

Combine the two integrated terms and add the constant of integration \(C\):
\(\int(4x^{-5}+8)dx=-x^{-4}+8x + C\)

Answer:

\(-x^{-4}+8x + C\)