If `int \ sqrt(cos^3x/sin^11x) \ dx = -2(A tan^(9//2)x + B tan^(5//2) x) + C,` then find A and B.

Options

A. `A=(1)/(9),B=-(1)/(5)`
B. `A=(1)/(9),B=(1)/(5)`
C. `A=-(1)/(9),B=(1)/(5)`
D. none of these

Answer

B

Explanation

We observe that the sum of the exponents of cos x and sin x is `((3)/(4)-(11)/(2))=-4=a` negative even integer.
So , we divide numberator and denominatior by `cos^(4)x`
`becauseI=intsqrt((cos^(3)x)/(sin^(11)x))dx=int(cos^(3//2)x)/(sin^(11//2)x)dx`
`rArrI=int(cos^(3//2)x xxcos^(4)x)/(sin^(11//2)x)xx(1)/(cos^(4)x)dx`
`rArrI=int(1)/(tan^(11//2)x)(1+tan^(2)x)d(tanx)`
`rArrI=int(tan^(-11//2)x+tan^(-7//2)x)d (tanx)`
`rArr=-(2)/(9)tan^(-9//2)x-(2)/(5)tan^(-5//2)x+C`
`rArr I=-{(1)/(9)tan^(-9//2)x+(1)/(5)tan^(-5//2)x}+C`
Hence , `A=(1)/(9)andB=(1)/(5)`