Integrate the function frac{cos x}{sqrt{1+sin | vedclass

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Question

(A) Let $1+\sin x = t$. Differentiating both sides with respect to $x$,we get $\cos x \, dx = dt$. Substituting these into the integral: $\int \frac{\

Vedclass · Accepted Answer

$2\sqrt{1+\sin x} + C$

Vedclass · Answer

(A) Let $1+\sin x = t$. Differentiating both sides with respect to $x$,we get $\cos x \, dx = dt$. Substituting these into the integral: $\int \frac{\cos x}{\sqrt{1+\sin x}} \, dx = \int \frac{dt}{\sqrt{t}} = \int t^{-\frac{1}{2}} \, dt$. Using the power rule $\int t^n \, dt = \frac{t^{n+1}}{n+1} + C$: $= \frac{t^{\frac{1}{2}}}{\frac{1}{2}} + C = 2\sqrt{t} + C$. Substituting back $t = 1+\sin x$: $= 2\sqrt{1+\sin x} + C$,where $C$ is an arbitrary constant.

Integrate the function $\frac{\cos x}{\sqrt{1+\sin x}}$.

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