# Evaluate this integral

1. Jun 21, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
$\displaystyle \int^1_0 cot^{-1}(x^2 - x +1)\ dx$

2. Relevant equations

3. The attempt at a solution
I used this formula

$2I=\int^b_a f(x)+f(a+b-x)\ dx$

But using this method I arrived at the original question. OK, So I tried integrating by parts and it's still useless. Substitution doesn't work either.

2. Jun 21, 2013

### tiny-tim

hi utkarshakash!

have you tried integrating by parts with u = x ?

3. Jun 21, 2013

### Pranav-Arora

$$\cot^{-1}(x^2-x+1)=\tan^{-1}\left(\frac{1}{x^2-x+1}\right)=\tan^{-1}\left(\frac{1}{1-x(1-x)}\right)$$

4. Jun 21, 2013

### haruspex

In case Pranav-Arora's hint is still obscure, compare it with the expansion of tan(a+b).

5. Jun 22, 2013

### sankalpmittal

Firstly write it as,

tan-1(1/(x2-x+1) = tan-1{(x-(x-1))/(1+x(x-1))}

Don't you see an obvious relation of tan-1A - tan-1B formula from here ? You may proceed..

It was an easy question though.

Method II :

Let f(x) =cot−1(x2−x+1)

Write as,

f(x) =cot−1(x2−x+1)*1

Take cot−1(x2−x+1) as a first function as per ILATE, and 1 as second function. Then you can integrate by parts.