Question

Integrate the following with respect to x:

${\displaystyle {(x+4)}^5+\frac{5}{{(2-5x)}^4}-{\text{cosec}}^2(3x-1)}$

Answer 1

${\displaystyle \int [{(x+4)}^5+\frac{5}{{(2-5x)}^4}-{\text{cosec}}^2(3x-1)]\text{d}x}$

= ${\displaystyle \int {(x+4)}^5\text{d}x+5\int \frac{1}{{(2-5x)}^4} \text{d}x-\int {\text{cosec}}^2(3x-1) \text{d}x}$

= ${\displaystyle \frac{{(x+4)}^6}{6}+\frac{-5}{-5\times 3{(2-5x)}^3}+\frac{\cot (3x-1)}{3}+\text{c}}$

= ${\displaystyle \frac{{(x+4)}^6}{6}+\frac{1}{3{(2-5x)}^3}+\frac{\cot (3x-1)}{3}+\text{c}}$