Question

1. \\( \int ( 2 x ^ { 5 } + 5 ) ^ { 4 } \cdot 10 x ^ { 4 } d x \\)

Explanation:

Step1: Substitute variable

Let \( u = 2x^{5}+5 \), then \( du=(2x^{5}+5)'dx = 10x^{4}dx \)

Step2: Integrate with respect to \( u \)

The integral \( \int(2x^{5}+5)^{4}\cdot10x^{4}dx=\int u^{4}du \)
Using the power - rule for integration \( \int u^{n}du=\frac{u^{n + 1}}{n+1}+C(n
eq - 1) \), here \( n = 4 \), so \( \int u^{4}du=\frac{u^{5}}{5}+C \)

Step3: Substitute back \( u = 2x^{5}+5 \)

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

Answer:

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