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Find the value of Integral

$\int \dfrac{sin \dfrac{5x}{2} }{sin \dfrac{x}{2} }$
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Answer 1

Answer :-))

x + 2sinx + sin2x + C

Solution:-)

$\int \dfrac{sin \dfrac{5x}{2} }{sin \dfrac{x}{2} }$

Multiply Numerator and Denominator by

$cos \dfrac{x}{2}$

$\int \dfrac{sin \dfrac{5x}{2}.cos \dfrac{x}{2} }{sin \dfrac{x}{2}. cos \dfrac{x}{2} }dx \\ \\ = \int \dfrac{sin{3x} + sin2x}{sinx}dx \\ \\ = \int \frac{(3sinx - 4sin {}^{3}x) + 2sinx.cosx}{sinx} dx \\ \\ = \int (3 - 4 {sin}^{2} x + 2cosx)dx \\ \\ \int(3 - 2(1 - cos2x + 2cosx))dx \\ \\ = \int(1 + cos2x + 2cosx)dx \\ \\ = x + 2sinx + sin2x + C$

where C is the constant of integration

Answer 2

Answer:

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