Question:
$\int\sqrt{1+\cos\,x}\,dx$ is equal to
$\int\sqrt{1+\cos\,x}\,dx$ is equal to
Updated On: Apr 26, 2024
- $2\sqrt{2} \cos \frac{x}{2}+C$
- $2\sqrt{2} \sin \frac{x}{2}+C$
- $\sqrt{2} \cos \frac{x}{2}+C$
- $\sqrt{2} \sin \frac{x}{2}+C$
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The Correct Option is B
Solution and Explanation
$\int \sqrt{1+\cos x} \,d x=\sqrt{2} \int \cos \left(\frac{x}{2}\right) d x$
$=2 \sqrt{2} \sin \left(\frac{x}{2}\right)+c$
$\left[\because 1+\cos x=2 \cos ^{2} \frac{x}{2}\right]$
$=2 \sqrt{2} \sin \left(\frac{x}{2}\right)+c$
$\left[\because 1+\cos x=2 \cos ^{2} \frac{x}{2}\right]$
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
