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evaluate the following integral ∫ sin(x – alpha) / sin(x + alpha) dx

evaluate the following integral
∫ sin(x – alpha) / sin(x + alpha) dx

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:
Hello student,
Please find the answer to your question below

I = \int \frac{sin(x-\alpha )}{sin(x+\alpha )}dx
I = \int \frac{sin((x+\alpha) -2\alpha )}{sin(x+\alpha )}dx
I = \int \frac{sin(x+\alpha) cos2\alpha-cos(x+\alpha )sin2\alpha }{sin(x+\alpha )}dx
I = \int cos2\alpha dx-\int \frac{cos(x+\alpha )sin2\alpha }{sin(x+\alpha )}dx
I = (cos2\alpha )x-sin2\alpha \int \frac{cos(x+\alpha )}{sin(x+\alpha )}dx
sin(x+\alpha ) = t
cos(x+\alpha )dx = dt
I = (cos2\alpha )x-sin2\alpha \int \frac{dt}{t}
I = (cos2\alpha )x-sin2\alpha ln(t)+c
I = (cos2\alpha )x-sin2\alpha ln(sin(x+\alpha ))+c

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