Question

$\mathrm{If}\int \left(\sqrt{1+\sin x}\right)f\left(x\right)\mathrm{dx}=\frac{2}{3}{(1+\sin x)}^{3/2}+C,\mathrm{then}f\left(x\right)\mathrm{is}$
.

• A. cos x
• B. sin x
• C. tan x
• D. 1

Answer 1

The correct option is A cos x

Here, the important thing to remember is that integration is the reverse process of differentiation i.e.
If
$\int g\left(x\right)\mathrm{dx}=F\left(x\right)+C$
, then
$\frac{\mathrm{dF}\left(x\right)}{\mathrm{dx}}=g\left(x\right)$.
$\mathrm{Here},\mathrm{let}\int g\left(x\right)\mathrm{dx}=\int \left(\sqrt{1+\mathrm{sinx}}\right)f\left(x\right)\mathrm{dx}\Rightarrow g\left(x\right)=\left(\sqrt{1+\mathrm{sinx}}\right)f\left(x\right)\mathrm{also}F\left(x\right)=\frac{2}{3}{(1+\mathrm{sinx})}^{3/2}+C\Rightarrow \int g\left(x\right)=F\left(x\right)\Rightarrow g\left(x\right)=\frac{\mathrm{dF}\left(x\right)}{\mathrm{dx}}$

$\mathrm{Thus},\mathrm{substituting}\mathrm{the}\mathrm{function}\mathrm{values}\mathrm{in}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{we}\mathrm{get},\frac{d}{\mathrm{dx}}\left(\frac{2}{3}{(1+\mathrm{sinx})}^{3/2}+C\right)=f\left(x\right)\sqrt{1+\mathrm{sinx}}\Rightarrow \frac{2}{3}.\frac{3}{2}{(1+\mathrm{sinx})}^{1/2}\cos x=f\left(x\right)\sqrt{1+\mathrm{sinx}}\Rightarrow {(1+\mathrm{sinx})}^{1/2}\cos x=f\left(x\right)\sqrt{1+\mathrm{sinx}}\Rightarrow \cos x=f\left(x\right)$