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int("sin"(5x)/(2))/("sin"(x)/(2))dx is e...

`int("sin"(5x)/(2))/("sin"(x)/(2))dx` is equal to (where, C is a constant of integration)

A

`2x + "sin" x + 2 "sin" 2x + C`

B

`x + 2"sin" x + 2 "sin" 2x + C`

C

`x + 2"sin" x + "sin" 2x + C`

D

`2x + "sin" x + "sin" 2x + C`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `I=int("sin"(5x)/(2))/("sin"(x)/(2))dx = int("2 sin"(5x)/(2)"cos"(x)/(2))/("2 sin"(x)/(2)"cos"(x)/(2))dx` [multiplying by `"2 cos"(x)/(2)` in numerator and denominator]
`=int(sin3x + sin2x)/(sin x)dx`
`[therefore 2 sin A cos B = sin(A+B)+sin(A-B)and sin 2 A = 2 sin A cos A]`
`= int((3 sin x - 4 sin^(3) x)+2 sin x cos x)/(sin x)dx" "[therefore sin 3x = 3 sin x -4 sin^(3) x]`
`= int(3-4 sin^(2)x + 2 cos x)dx`
`=int[3-2(1-cos 2x)+2 cos x]dx" "[therefore 2 sin^(2)x=1 - cos 2x]`
`= int[3-2+2 cos 2x + 2 cos x]dx`
`=[1+2 cos 2x + 2 cos x]dx`
`=x+2 sin x + sin 2x + C`
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