Let d/(dx)F(x)=(e^(sinx))/x . x lt0. If ...

Let `d/(dx)F(x)=(e^(sinx))/x . x lt0`. If `int_(1)^(4)(2e^(sinx^(2)))/xdx=F(k)-F(1)` then find the possible value of `k`.

A

4

B

8

C

16

D

32

Text Solution

Verified by Experts

The correct Answer is:

C

We have,
`(d)/(dx)(f(x))=(e^(sinx))/(x)rArr int(e^(sinx))/(x)dx=F(x) " "`.......(i)
Now,
`overset(4)underset(1)int 2e^(sinx^(2))/(x)dx=overset(4)underset(1)int (e^(sinx^(2)))/(x^(2)).d(x^(2))=overset(16)underset(1)int (e^(sin t))/(t)dt,"where" t=x^(2)`
`rArr overset(4)underset(1)int 2(e^(sinx^(2)))/(x)dx=[F(t)]_(1)^(16)=F(16)-F(1)" "`[Using (i)]
Hence, k=16

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