Question

1. ∫100x⁴/(5x⁵ - 4) dx

Explanation:

Step1: Let \( u = 5x^{5}-4 \)

Differentiate \( u \) with respect to \( x \): \( du=25x^{4}dx \), so \( 4du = 100x^{4}dx \)

Step2: Substitute into the integral

The integral \( \int\frac{100x^{4}}{5x^{5}-4}dx \) becomes \( \int\frac{4du}{u} \)

Step3: Integrate

Using the formula \( \int\frac{1}{u}du=\ln|u| + C \), we get \( 4\ln|u|+C \)

Step4: Substitute back \( u = 5x^{5}-4 \)

The result is \( 4\ln|5x^{5}-4|+C \)

Answer:

\( 4\ln|5x^{5}-4|+C \)