Question

evaluate.
\\( \int \frac{5x^4}{(6 + x^5)^4} \\, dx \\)
\\( -\frac{5x^4}{(6 + x^5)^3} + c \\)
\\( -\frac{1}{5(6 + x^5)^5} + c \\)
\\( -(6 + x^5)^5 + c \\)
\\( -\frac{1}{3(6 + x^5)^3} + c \\)

Explanation:

Step1: Use Substitution Method

Let \( u = 6 + x^5 \). Then, find the derivative of \( u \) with respect to \( x \).
\( \frac{du}{dx} = 5x^4 \), so \( du = 5x^4 dx \).

Step2: Rewrite the Integral

Substitute \( u \) and \( du \) into the integral:
\( \int \frac{5x^4}{(6 + x^5)^4} dx = \int \frac{du}{u^4} \)

Step3: Integrate with Power Rule

Recall the power rule for integration: \( \int u^n du = \frac{u^{n + 1}}{n + 1} + C \) (for \( n
eq -1 \)). Here, \( n = -4 \).
\( \int u^{-4} du = \frac{u^{-4 + 1}}{-4 + 1} + C = \frac{u^{-3}}{-3} + C = -\frac{1}{3u^3} + C \)

Step4: Substitute Back \( u \)

Replace \( u \) with \( 6 + x^5 \):
\( -\frac{1}{3(6 + x^5)^3} + C \)

Answer:

\( -\frac{1}{3(6 + x^5)^3} + C \) (the fourth option: \( -\frac{1}{3(6 + x^5)^3} + C \))