यदि `f(x)=int_(1//x)^(sqrt(x))cost^(2)dt,xgt0`
तथा `f'(x)=(1)/(x^(2))A+(1)/(2)B`
B का मान ज्ञात कीजिए ।

Options

A. `(1)/(sqrt(x))sinx`
B. `(1)/(sqrt(x))cosx`
C. `(1)/(sqrt(2))sinx`
D. `(1)/(2)cosx`

Answer

B

Explanation

`f(x)=int_(1//x)^(sqrt(x))cost^(2)dt=int_(1//x)^(0)cost^(2)dt+int_(0)^(sqrt(x))cost^(2)dt`
`=-int_(0)^(1//x)cost^(2)dt+int_(0)^(sqrt(x))cost^(2)dt`
`thereforef'(x)=-[cost^(2)]_(t=(1)/(x))`
`(d)/(dx)((1)/(x))+[cost^(2)]_(t=sqrt(x))(d)/(dx)(sqrt(x))`
`=(1)/(x^(2))"cos"(1)/(x^(2))+(1)/(2sqrt(x))cosx=(1)/(x^(2))A+(1)/(2)B`
`B=(1)/(sqrt(x))cosx`