Question

$\int 2x\left(x^{2}+5\
ight)^{3}dx$

Explanation:

Step1: Choose substitution

Let \( u = x^2 + 5 \), then \( du = 2x \, dx \).

Step2: Substitute into integral

The integral becomes \( \int u^3 \, du \).

Step3: Integrate using power rule

Using the power rule for integration \( \int u^n du=\frac{u^{n + 1}}{n+1}+C\) (where \( n = 3\)), we get \( \frac{u^{4}}{4}+C \).

Step4: Substitute back \( u \)

Substitute \( u = x^2 + 5 \) back into the expression, we have \( \frac{(x^2 + 5)^{4}}{4}+C \).

Answer:

\(\frac{(x^2 + 5)^{4}}{4}+C\)