$\int\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx $ is equal to :
(where $c$ is a constant of integration)
- $2x + sin \,x + 2 sin\,2x + c$
- $x + 2\,sinx + 2\,sin2x + c$
- $x + 2\, sin x + sin \,2x + c$
- $2x + sinx + sin2x + c$
The Correct Option is C
Solution and Explanation
$ =\int \frac{\sin 3x +\sin 2x}{\sin x} dx $
$ = \int \frac{3\sin x-4\sin^{3}x-2\sin x\cos x}{\sin x}dx $
$ = \int \left(3-4\sin^{2}x+2\cos x \right)dx $
$ =\int \left(3-2\left(1-\cos2x\right)+2\cos x\right)dx $
$ =\int\left(1+2\cos2x+2\cos x\right)dx $
$ =x+\sin2x+2\sin x +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.
