Question

If $f\left(x\right)=\int \frac{dx}{{\sin }^{1/2}x{\sin }^{7/2}x}$, then $f\left(\frac{\pi }{4}\right)-f\left(0\right)=$

• A. $2.67$
• B. $2.3$
• C. $2.4$
• D. $2.5$

Answer 1

The correct option is A $2.67$

Mistake :$\cos x$ must be there instead of $\sin x$

$\int \frac{dx}{{\sin }^{1/2}x{\cos }^{7/2}x}=\int \frac{{\text{sec}}^{4}x}{{\tan }^{1/2}x}dx=\int ({\tan }^{-1/2}x+{\tan }^{3/2}x){\text{sec}}^{2}xdx=2{\tan }^{1/2}x+\frac{{\tan }^{3/2}x}{\frac{3}{2}}+C$

$f\left(\frac{\pi }{4}\right)-f\left(0\right)=2+\frac{2}{3}=\frac{8}{3}=2.67$