∫ 1 + cos x d x equals

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Question: Answered & Verified by Expert

$\int \sqrt{1+\cos x} d x$ equals

MathematicsIndefinite IntegrationJEE Main

Options:

A $2 \sqrt{2} \sin \frac{x}{2}+c$
B $-2 \sqrt{2} \sin \frac{x}{2}+c$
C $-2 \sqrt{2} \cos \frac{x}{2}+c$
D $2 \sqrt{2} \cos \frac{x}{2}+c$

Solution:

1556 Upvotes Verified Answer

The correct answer is: $2 \sqrt{2} \sin \frac{x}{2}+c$

$\begin{aligned} & I=\int \sqrt{1+\cos x} d x=\int \sqrt{2 \cos ^2(x / 2)} d x \\ & I=\sqrt{2} \int \cos (x / 2) d x=2 \sqrt{2} \sin (x / 2)+c\end{aligned}$

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