Indefinite Integrals — Maths STD 12 Science — Question

Let $\sin\text{x}=\text{t}$

$\Rightarrow\cos\text{x}\text{ dx}=\text{dt}$

Now, $\int\frac{(1-\sin^4\text{x}-2\sin^2\text{x})}{\sin\text{x}}\cos\text{x}\text{ dx}$

$=\int\frac{(1+\text{t}^4-2\text{t}^2)}{\text{t}}\text{ dt}$

$=\int\Big(\frac{1}{\text{t}}+\text{t}^3-2\text{t}\Big)\text{dt}$

$=\log|\text{t}|+\frac{\text{t}^4}{4}-\frac{2\text{t}^2}{2}+\text{C}$

$=\log|\text{t}|+\frac{\text{t}^4}{4}-\text{t}^2+\text{C}$

$=\log|\sin\text{x}|+\frac{\sin^4\text{x}}{4}-\sin^2\text{x}+\text{C}$