If [ ^(3)/(2) x+ ^(3)/(2) x ^3 x ^3 x (x-) d x=A x- +B - x+C], where C is the integration …

Question

If $$\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C$$, where $C$ is the integration constant, then $A B$ is equal to

Accepted Answer

$$\begin{aligned}
& \text {} \int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x \
& I=\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x(\sin x \cos \theta-\cos x \sin \theta)}} d x \
& =\int \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x \cos ^2 x \sqrt{\tan x \cos \theta-\sin \theta}} d x+\int \frac{\cos ^{\frac{3}{2}} x}{\sin ^2 x \cos ^{\frac{3}{2}} x \sqrt{\cos \theta-\cot x \sin \theta}} d x= \
& \int \frac{\sec ^2 x}{\sqrt{\tan x \cos \theta-\sin \theta}} d x+\int \frac{\operatorname{cosec}^2 x}{\sqrt{\cos \theta-\cot x \sin \theta}} d x \
&
\end{aligned}$$

$$\mathrm{I}=\mathrm{I}_1+\mathrm{I}_2$$ ...... {Let}

For $$\mathrm{I}_1$$, let $$\tan \mathrm{x} \cos \theta-\sin \theta=\mathrm{t}^2$$

$$\sec ^2 x d x=\frac{2 t d t}{\cos \theta}$$

For $$\mathrm{I}_2$$, let $$\cos \theta-\cot \mathrm{x} \sin \theta=\mathrm{z}^2$$

$$\operatorname{cosec}^2 x d x=\frac{2 z d z}{\sin \theta}$$

$$\begin{aligned}
& I=I_1+I_2 \
& =\int \frac{2 t d t}{\cos \theta t}+\int \frac{2 z d z}{\sin \theta z} \
& =\frac{2 t}{\cos \theta}+\frac{2 z}{\sin \theta} \
& =2 \sec \theta \sqrt{\tan x \cos \theta-\sin \theta}+2 \operatorname{cosec} \theta \sqrt{\cos \theta-\cot x \sin \theta} \
& \text { Comparing } \
& \quad A B=8 \operatorname{cosec} 2 \theta
\end{aligned}$$

Solutions · 1