current objective use substitution to evaluate an indefinite integral with the power rule question evaluate…

Explanation:

Step1: Set substitution

Let $u = 5-3x^{4}$. Then $du=-12x^{3}dx$, and $x^{3}dx=-\frac{1}{12}du$.

Step2: Rewrite the integral

The original integral $\int\frac{5x^{3}}{(5 - 3x^{4})^{8}}dx$ becomes $\int\frac{5}{u^{8}}\cdot(-\frac{1}{12})du=-\frac{5}{12}\int u^{- 8}du$.

Step3: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $n=-8$, we have $-\frac{5}{12}\int u^{-8}du=-\frac{5}{12}\cdot\frac{u^{-8 + 1}}{-8+1}+C$.

Step4: Simplify the result

$-\frac{5}{12}\cdot\frac{u^{-7}}{-7}+C=\frac{5}{84u^{7}}+C$.

Step5: Substitute back $u$

Substitute $u = 5-3x^{4}$ back into the result, we get $\frac{5}{84(5 - 3x^{4})^{7}}+C$.

Answer:

$\frac{5}{84(5 - 3x^{4})^{7}}+C$