Question:
$ \int_{0}^{x}{\log \,(\cot \,x\,+\,\tan t)\,dt} $ =
$ \int_{0}^{x}{\log \,(\cot \,x\,+\,\tan t)\,dt} $ =
Updated On: Jul 27, 2024
- $ x\,\log \,(\sin \,x) $
- $ -x\,\log \,(\sin \,x) $
- $ x\,\log \,(cos\,x) $
- $ -x\,\log \,(cos\,x) $
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The Correct Option is B
Solution and Explanation
Let $ I=\int_{0}^{x}{\log \,(\cot \,x+\tan t)\,dt} $
$ =\int_{0}^{x}{\log \left( \frac{\cos x}{\sin x}+\frac{\sin t}{\cos t} \right)dt} $
$ =\int_{0}^{x}{[\log \,\{\cos \,(x-t)\}-\log \,\sin \,x-\log \,\cos t]dt} $
$ =\int_{0}^{x}{\log \,\{\cos (x-x+t)\}\,dt} $ $ -\int_{0}^{x}{\log \,\,\sin x\,dt-\int_{0}^{x}{\log \cos \,tdt}} $
$ =\int_{0}^{x}{\log \,\cos t\,dt-[t\,\log \,\sin \,x]_{0}^{x}-\int_{0}^{x}{\log \,\,\cos \,t\,\,dt}} $
$ =-(x\,\log \,\sin x) $
$ =\int_{0}^{x}{\log \left( \frac{\cos x}{\sin x}+\frac{\sin t}{\cos t} \right)dt} $
$ =\int_{0}^{x}{[\log \,\{\cos \,(x-t)\}-\log \,\sin \,x-\log \,\cos t]dt} $
$ =\int_{0}^{x}{\log \,\{\cos (x-x+t)\}\,dt} $ $ -\int_{0}^{x}{\log \,\,\sin x\,dt-\int_{0}^{x}{\log \cos \,tdt}} $
$ =\int_{0}^{x}{\log \,\cos t\,dt-[t\,\log \,\sin \,x]_{0}^{x}-\int_{0}^{x}{\log \,\,\cos \,t\,\,dt}} $
$ =-(x\,\log \,\sin x) $
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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.
