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int (0)^(1) (xdx)/([x + sqrt(1-x^(2))]sq...

` int _(0)^(1) (xdx)/([x + sqrt(1-x^(2))]sqrt(1-x^(2)))` is equal to

A

0

B

1

C

`pi/4`

D

`(pi^(2))/2`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `l=int_(0)^(1)("x dx")/([x + sqrt(1-x^(2))]sqrt(1-x^(2)))`
Put `x = sin theta rArr dx = cos theta d theta`
`l = int_(0)^(pi//2)(sin theta. Cos theta d theta)/((sin theta + cos theta).cos theta)`
`rArr l = int_(0)^(pi//2)(sin theta)/(sin theta + cos theta)` ...(i)
`rArr l = int_(0)^(pi//2)(sin((pi)/(2)-theta))/(sin((pi)/(2)-theta)+cos((pi)/(2)-theta))d theta`
`rArr l = int_(0)^(pi//2)(cos theta)/(sin theta + cos theta)d theta` ...(ii)
On adding Eqs. (i) and (ii), we get
`2l = int_(0)^(pi//2)((sin theta + cos theta)/(sin theta + cos theta))d theta`
`= int_(1)^(pi//2)1 d theta = (pi)/(2) rArr l = (pi)/(4)`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  11. The value of overset(pi)underset(-pi)int(1-x^(2)) sin x cos^(2) x" dx"...

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