Question:
$ \int{\frac{\sec x cosec x}{2\cot x-\sec x\cos ecx}}dx $ is equal to
$ \int{\frac{\sec x cosec x}{2\cot x-\sec x\cos ecx}}dx $ is equal to
Updated On: Jun 6, 2022
- $ \log |\sec x+\tan x|+c $
- $ \log |\sec x+\cos ecx|+c $
- $ \frac{1}{2}\log |\sec 2x+\tan 2x|+c $
- $ \log |\sec 2x+\cos ec2x|+c $
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The Correct Option is C
Solution and Explanation
$ \int{\frac{\sec x\cos ecx}{2\cot x-\sec x\cos ecx}}dx $
$=\int{\frac{\frac{1}{\cos x\sin x}}{\frac{2\cos x}{\sin x}-\frac{1}{\sin x\cos x}}}dx $
$=\int{\frac{dx}{2{{\cos }^{2}}x-1}} $
$=\int{\frac{dx}{{{\cos }^{2}}x-{{\sin }^{2}}x}=\int{\sec 2x\,dx}} $
$=\frac{1}{2}\log |\sec 2x+\tan 2x|+c $
$=\int{\frac{\frac{1}{\cos x\sin x}}{\frac{2\cos x}{\sin x}-\frac{1}{\sin x\cos x}}}dx $
$=\int{\frac{dx}{2{{\cos }^{2}}x-1}} $
$=\int{\frac{dx}{{{\cos }^{2}}x-{{\sin }^{2}}x}=\int{\sec 2x\,dx}} $
$=\frac{1}{2}\log |\sec 2x+\tan 2x|+c $
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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.
