Find the indefinite integral.
\[
\int \frac{x^{4}}{\left(9-x^{5}\right)^{2}} d x
\]

Answer

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Answer

Simplify the expression to get the final answer, \(\boxed{\frac{1}{9-x^{5}} + C}\).

Steps

Step 1 ：First, we recognize that the integral is in the form of a standard integral, which is \(\int \frac{f'(x)}{[f(x)]^n} dx = -\frac{1}{(n-1)f(x)^{n-1}} + C\), where \(f(x) = 9 - x^5\), \(f'(x) = -5x^4\), and \(n = 2\).

Step 2 ：We can rewrite the integral as \(\int \frac{-(-5x^{4})}{(9-x^{5})^{2}} dx\).

Step 3 ：Now, we can apply the formula, where \(f'(x) = -5x^4\), \(f(x) = 9 - x^5\), and \(n = 2\).

Step 4 ：So, the integral becomes \(-\frac{1}{(2-1)(9-x^{5})^{2-1}} + C\).

Step 5 ：Simplify the expression to get the final answer, \(\boxed{\frac{1}{9-x^{5}} + C}\).

Find the indefinite integral.\[\int \frac{x^{4}}{\left(9-x^{5}\right)^{2}} d x\]

Answer

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Find the indefinite integral.
\[
\int \frac{x^{4}}{\left(9-x^{5}\right)^{2}} d x
\]