Question

question 15 - 1 point evaluate the indefinite integral given below. provide your answer below. (int\frac{4x^{2}}{(1 - 5x^{3})^{2}}dx=square)

Explanation:

Step1: Use substitution method

Let $u = 1 - 5x^{3}$, then $du=-15x^{2}dx$ and $x^{2}dx=-\frac{1}{15}du$. The integral $\int\frac{4x^{2}}{(1 - 5x^{3})^{2}}dx$ becomes $\int\frac{4}{u^{2}}\cdot(-\frac{1}{15})du=-\frac{4}{15}\int u^{- 2}du$.

Step2: Integrate $u^{-2}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), for $n=-2$, we have $\int u^{-2}du=\frac{u^{-2 + 1}}{-2+1}+C=-\frac{1}{u}+C$.

Step3: Substitute back $u = 1 - 5x^{3}$

$-\frac{4}{15}\int u^{-2}du=-\frac{4}{15}\cdot(-\frac{1}{u})+C=\frac{4}{15(1 - 5x^{3})}+C$.

Answer:

$\frac{4}{15(1 - 5x^{3})}+C$