Question:
$\int \frac{dx}{\sin x - \cos x + \sqrt{2}} $ equals to
$\int \frac{dx}{\sin x - \cos x + \sqrt{2}} $ equals to
Updated On: Apr 15, 2024
- $ - \frac{1}{\sqrt{2}} \tan \left(\frac{x}{2} + \frac{\pi}{8}\right) + C $
- $ \frac{1}{2} \tan \left(\frac{x}{2} + \frac{\pi}{8}\right) + C $
- $ \frac{1}{\sqrt{2}} \cot\left(\frac{x}{2} + \frac{\pi}{8}\right) + C $
- $ - \frac{1}{\sqrt{2}} \cot \left(\frac{x}{2} + \frac{\pi}{8}\right) + C $
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The Correct Option is D
Solution and Explanation
Let $I = \int \frac{dx}{\sin x - \cos x + \sqrt{2}} $
$= \int \frac{dx}{\sqrt{2}\left(\sin x \sin \frac{\pi}{4} - \cos x \cos \frac{\pi}{4} + 1\right)} $
$= \frac{1}{\sqrt{2}}\int \frac{dx}{1-\cos\left(x + \frac{\pi}{4}\right)} $
$= \frac{1}{\sqrt{2}}\int \frac{dx}{2 \sin^{2} \left(\frac{x}{2} + \frac{\pi }{8}\right)} $
$= \frac{1}{2\sqrt{2}}\int cosec^{2}\left(\frac{x}{2} + \frac{\pi }{8}\right)dx$
$ = \frac{1}{2\sqrt{2}} \frac{-\cot \left(\frac{x}{2}+ \frac{\pi }{8}\right)}{\frac{1}{2}} +C$
$ = -\frac{1}{\sqrt{2}}\cot\left(\frac{x}{2}+\frac{\pi}{8}\right)+C $
$= \int \frac{dx}{\sqrt{2}\left(\sin x \sin \frac{\pi}{4} - \cos x \cos \frac{\pi}{4} + 1\right)} $
$= \frac{1}{\sqrt{2}}\int \frac{dx}{1-\cos\left(x + \frac{\pi}{4}\right)} $
$= \frac{1}{\sqrt{2}}\int \frac{dx}{2 \sin^{2} \left(\frac{x}{2} + \frac{\pi }{8}\right)} $
$= \frac{1}{2\sqrt{2}}\int cosec^{2}\left(\frac{x}{2} + \frac{\pi }{8}\right)dx$
$ = \frac{1}{2\sqrt{2}} \frac{-\cot \left(\frac{x}{2}+ \frac{\pi }{8}\right)}{\frac{1}{2}} +C$
$ = -\frac{1}{\sqrt{2}}\cot\left(\frac{x}{2}+\frac{\pi}{8}\right)+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.
