Question

integration of 1/√sin^3 x. cos(x-a)

Answer 1

Given : 1/√sin³x. cos(x-a)

To Find : Integrate

Solution:

I =∫ 1/√sin³x. cos(x-a) dx

1/√sin³x. cos(x-a)

= 1/√sin³x. (cosxcosa + sinxsina)

= 1/√sin³x. sinx(cotxcosa + sina)

= 1/√sin⁴x(cotxcosa + sina)

= 1/sin²x.√(cotxcosa + sina)

=  cosec²x/√(cotxcosa + sina)

I = ∫(cosec²x/√(cotxcosa + sina)) dx

√(cotxcosa + sina)  = t

=> (1/2√(cotxcosa + sina))(-cosec²xcosa) dx = dt

=> (cosec²x/√(cotxcosa + sina)) dx = -2dt/cosa

I = ∫-2dt/cosa

=> I = - (2/cosa)∫dt

=>  I = - (2/cosa)t  + C

=> I  = - 2t/Cosa + C

substitute value of t

=> I  = - 2√(cotxcosa + sina) / cosa  + C

√(cotxcosa + sina)  = √(cos(x -a)/sinx)

Learn More:

Integrate dx/ 3sin^2x+5cos^2x - Brainly.in

https://brainly.in/question/9262036

Integration of x²×sec²(x³) dx - Brainly.in

https://brainly.in/question/18063473