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Only Champs Can AnswerLet's Go Show Me who is the Champion .Find the value of Integral​

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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


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