Question

find the indefinite integral and check the result by differentiation. (remember the constant of integration.)
int \frac { 1 } { x ^ { 6 } } d x

Explanation:

Step1: Rewrite the integrand

Rewrite \(\frac{1}{x^{6}}\) as \(x^{-6}\). So the integral becomes \(\int x^{-6}dx\).

Step2: Apply the power rule for integration

The power rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). Here \(n=-6\), then \(\int x^{-6}dx=\frac{x^{-6 + 1}}{-6+1}+C=\frac{x^{-5}}{-5}+C=-\frac{1}{5x^{5}}+C\).

Step3: Check by differentiation

Differentiate \(y =-\frac{1}{5x^{5}}+C=-\frac{1}{5}x^{-5}+C\) using the power rule for differentiation \((x^{n})^\prime=nx^{n - 1}\).
\(y^\prime=-\frac{1}{5}\times(-5)x^{-5-1}=x^{-6}=\frac{1}{x^{6}}\).

Answer:

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