Question

Assertion :Let F(x) be an indefinite integral of $si{n}^{-3/2}xco{s}^{-5/2}x$.
Statement 1: F is periodic function with period $2\pi$. Reason: Statement 2: The period of $si{n}^{n}x$ and $co{s}^{n}x$ is $2\pi$.

• A. Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation for Statement 1
• B. Both Statement 1 and Statement 2 are correct but Statement 2 is not the correct explanation for Statement 1
• C. Statement 1 is correct but Statement 2 is incorrect
• D. Both Statement 1 and Statement 2 are incorrect

Answer 1

The correct option is C Statement 1 is correct but Statement 2 is incorrect

Let $I=\int \frac{1}{\sqrt{{\sin }^{3}x{\cos }^{2}x}}dx$
Multiply numerator and denominator by ${\sec }^{4}x$
$I=\int \frac{{\sec }^{4}x}{{\tan }^{\frac{3}{2}}x}dx=\int \frac{(1+{\tan }^{2}x){\sec }^{2}x}{{\tan }^{\frac{3}{2}}x}dx$
Put $t=\tan x\Rightarrow dt={\sec }^{2}xdx$
$I=\int \frac{t^2+1}{t^{\frac{3}{2}}}dt$
Put $u=\sqrt{t}\Rightarrow du=\frac{1}{2\sqrt{t}}dt$
$I=2\int \frac{u^4+1}{u^2}du=2\int (u^2+\frac{1}{u^2})du$
$=\frac{2u^3}{3}-\frac{2}{u}=\frac{2({\tan }^{2}x-3)}{2\sqrt{\tan x}}$
$=\frac{-\sin 2x-\sin 4x}{3\sqrt{{\sin }^{3}x{\cos }^{5}x}}$
Now for ${\sin }^{n}x,{\cos }^{n}x$
If n is even period is $\pi$
And if n is odd period is $2\pi$