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Ifint0^1cot^(-1)(1-x+x^2)dx=lambdaint0^1...

`Ifint_0^1cot^(-1)(1-x+x^2)dx=lambdaint_0^1tan^(-1)x dx ,then lambda is equal to` 1 (b) 2 (c) 3 (d) 4

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
B
`int_(0)^(1)cot^(-1)(1-x+x^(2))dx`
`=int_(0)^(1)tan^(-1)(1/(1-x+x^(2)))dx`
`=int_(0)^(1)tan^(-1)((x+(1-x))/(1-x(1-x)))dx`
`=int_(0)^(1)tan^(-1)x dx+int_(0)^(1)tan^(-1)(1-x)dx`
`=int_(0)^(1)tan^(-1)x dx+int_(0)^(1)tan^(-1)[1-(1-x)]dx`
`=2int_(0)^(1)tan^(-1)xdx` or `lamda=2`
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