find the indefinite integral.\n∫(x^5/(4 - x^6)^2)dx

Explanation:

Step1: Use substitution

Let $u = 4 - x^{6}$, then $du=-6x^{5}dx$, and $x^{5}dx=-\frac{1}{6}du$.

Step2: Rewrite the integral

The integral $\int\frac{x^{5}}{(4 - x^{6})^{2}}dx$ becomes $\int\frac{1}{u^{2}}\cdot(-\frac{1}{6})du=-\frac{1}{6}\int u^{- 2}du$.

Step3: Integrate using power - rule

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $n=-2$, we have $-\frac{1}{6}\int u^{-2}du=-\frac{1}{6}\cdot\frac{u^{-2 + 1}}{-2 + 1}+C$.

Step4: Simplify the result

$-\frac{1}{6}\cdot\frac{u^{-1}}{-1}+C=\frac{1}{6u}+C$.

Step5: Substitute back $u$

Substituting $u = 4 - x^{6}$ back, we get $\frac{1}{6(4 - x^{6})}+C$.

Answer:

$\frac{1}{6(4 - x^{6})}+C$