Question:
The integral $\int \sqrt{ 1 + 2 \cot \, x (cosec \, x + \cot \, x) } dx \, \left( 0 < x < \frac{\pi}{2} \right)$ is equal to :
(where $C$ is a constant of integration)
The integral $\int \sqrt{ 1 + 2 \cot \, x (cosec \, x + \cot \, x) } dx \, \left( 0 < x < \frac{\pi}{2} \right)$ is equal to :
(where $C$ is a constant of integration)
Updated On: Feb 14, 2025
- $4 \log\left(\sin \frac{x}{2}\right) + C $
- $2 \log\left(\sin \frac{x}{2}\right) + C $
- $2 \log\left(\cos\frac{x}{2}\right) + C $
- $4 \log\left(\cos\frac{x}{2}\right) + C $
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The Correct Option is B
Solution and Explanation
$\int\left(\sqrt{+2cot x cos ecx + cos ec^{2}x + cot x}\right)dx$
$\int cos | x + cot x | dx$
$\int\left(cos ec + cot x\right)dx$
$\int cosec dx$
$2log\left(log\left(x_{2} \right) + c\right)$
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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.
