Question

1. \\( \int ( 3 x ^ { 5 } + 4 ) ^ { 3 } \cdot 15 x ^ { 4 } d x \\)

Explanation:

Step1: Let \( u = 3x^{5}+4 \)

Differentiate \( u \) with respect to \( x \): \( du=(3\times5x^{4}+0)dx = 15x^{4}dx \)

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

The integral \( \int(3x^{5}+4)^{3}\cdot15x^{4}dx \) becomes \( \int u^{3}du \)

Step3: Integrate \( \int u^{3}du \)

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

Step4: Substitute back \( u = 3x^{5}+4 \)

We get \( \frac{(3x^{5}+4)^{4}}{4}+C \)

Answer:

\( \frac{(3x^{5}+4)^{4}}{4}+C \)