Question

1. $\int 2x(5x + 2)^{3}dx$

Explanation:

Step1: Expand the integrand

$$\begin{align*} (5x + 2)^3&=(5x)^3+3\times(5x)^2\times2 + 3\times(5x)\times2^2+2^3\\ &=125x^3 + 150x^2+60x + 8 \end{align*}$$

Then \(2x(5x + 2)^3=2x(125x^3 + 150x^2+60x + 8)=250x^4+300x^3 + 120x^2+16x\)

Step2: Integrate term - by - term

Use the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\)
\(\int(250x^4+300x^3 + 120x^2+16x)dx\)
\(=250\int x^4dx+300\int x^3dx+120\int x^2dx+16\int xdx\)
\(=250\times\frac{x^{5}}{5}+300\times\frac{x^{4}}{4}+120\times\frac{x^{3}}{3}+16\times\frac{x^{2}}{2}+C\)

Step3: Simplify the result

\(=50x^5 + 75x^4+40x^3+8x^2+C\)

Answer:

\(50x^5 + 75x^4+40x^3+8x^2+C\)