Question

integral of whole root of (1+Cos x dx)​

Answer 1

Answer:

refer the attachment below

Answer 2

Concept

Recall the formula which will use to solve the problem

1. $\cos x=2 \cos ^2x-1$

2. $\int \cos x= \sin x$

Given

The integral $(1+\cos x)^{\frac{1}{2}}$.

To find

We have to find the value.

Solution

First simplify the expression

$1+\cos x =1+2\cos^2 \frac{x}{2} -1$

              $=2\cos^2 \frac{x}{2}$

then intregrate with respect to x

$\int (1+\cos x)^{\frac{1}{2}}=\int \sqrt{2\cos^2 \frac{x}{2}}$

                     $=\int \sqrt{2} \cos \frac{x}{2}$

                     $=2\sqrt{2} \sin \frac{x}{2}$

Hence the final value is $2\sqrt{2} \sin x$.