Question

$\int_{}^{}\frac{1}{\sqrt{\sin^{3}x\sin(x + a)}}$ dx is equal to –

• A. – $\sqrt{\cos a + \cot x\sin a} + c$
• B. $\sqrt{\cos a + \cot x\sin a} + c$
• C. – $\sqrt{\cos a + \cot x\sin a} + c$
• D. $\sqrt{\cos a + \cot x\sin a} + c$

Answer 1

Answer: (A)

– $\sqrt{\cos a + \cot x\sin a} + c$

Answer 2

We have $\int_{}^{}\frac{1}{\sqrt{\sin^{3}x\sin(x + a)}}$dx

= $\int_{}^{}\frac{1}{\sqrt{\sin^{3}x(\sin x\cos a + \cos x\sin a)}}$dx

= $\int \frac { 1 } { \sqrt { \sin ^ { 4 } x ( \cos a + \cot x \sin a ) } }$dx

=$\int \frac { \operatorname { cosec } ^ { 2 } x } { \sqrt { \cos a + \cot x \sin a } }$dx

=$\int_{}^{}\frac{1}{\sqrt{\cos a + \cot x\sin a}}$d (cos a + cot x sin a)

=$\int_{}^{}\frac{1}{\sqrt{t}}$dt, where t = cos a + cot x sin a

= – $\frac{2}{\sin a}$ $\sqrt{\cos a + \cot x\sin a} + c$.

Hence (1) is the correct answer.