Question

int (6x + 5)(3x^{2}+5x)^{4}dx

Explanation:

Step1: Let \( u = 3x^{2}+5x \)

Differentiate \( u \) with respect to \( x \): \( du=(6x + 5)dx \)

Step2: Substitute \( u \) and \( du \) into the integral

The integral \( \int(6x + 5)(3x^{2}+5x)^{4}dx=\int u^{4}du \)

Step3: Integrate \( \int u^{4}du \)

Using the power - rule for integration \( \int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1) \), we have \( \int u^{4}du=\frac{u^{5}}{5}+C \)

Step4: Substitute back \( u = 3x^{2}+5x \)

The result is \( \frac{(3x^{2}+5x)^{5}}{5}+C \)

Answer:

\( \frac{(3x^{2}+5x)^{5}}{5}+C \)