The value of the constant `a gt 0` such that `int_(0)^(a) [tan^(-1)sqrt(x)]dx=int_(0)^(a) [cot^(-1)sqrt(x)]dx`, where[.] denotes the greatest integer function, is

Options

A. `(2(3+cos4))/(1-cos4)`
B. `(2(3-cos4))/(1+cos4)`
C. `(2(3-cos4))/(1-cos4)`
D. `(2(3+cos4))/(1+cos4)`

Answer

A

Explanation

We have,
`underset(0)overset(a)int [tan^(-1)sqrt(x)]dx=underset(0)overset(a)int [cot^(-1)sqrt(x)]dx`
`rArr underset(0)overset(tan^(2)1)int [tan^(-1)sqrt(x)]dx+underset(tan^(2)1)overset(a)int [tan^(-1)sqrt(x)]dx=underset(0)overset(cot^(2)1)int [cot^(-1)sqrt(x)]dx+underset(cot^(2)1)overset(a)int [cot^(-1)sqrt(x)]dx`
`rArr underset(0)overset(tan^(2)1)int 0 dx+underset(tan^(2)1)overset(a)int 1 dx=underset(0)overset(cot^(2)1)1.dx+underset(cot^(2)1)overset(a)int 0 dx`
`rArr a-tan^(2)1=cot^(2)1`
`rArr a=tan^(2)1+cot^(2)1=(sin^(4)1+cos^(2)1)/(sin^(2)1cos^(2)1)`
`rArr a=((sin^(2)1+cos^(2)1)^(2)-2sin^(2)1cot^(2)1)/((sin1 cos1)^(2))`
`a=(1-(1)/(2)sin^(2)2)/((1)/(2)sin^(2)2)=(4-2sin^(2)2)/(sin^(2)2)=(4-(1-cos4))/((1)/(2)(1-cos4))`
`rArr a=(2(3+cos4))/(1-cos4)`