# Homework Help: Challenging Integrals For Fun (1):

1. Feb 8, 2010

### Ratio Test =)

Hello :)
Here we go! :

1. $$\int \frac{dx}{x^3+x^2+x+1}$$

2. $$\int_{\frac{\pi}{2}}^{\pi} \left( sin(x) ln(x) - \frac{cos(x)}{x}\right) dx$$

2. Feb 8, 2010

### Count Iblis

1 + x + x^2 + x^3 = (1-x^4)/(1-x)

So, we have to integrate -(x-1)/(x^4 - 1)

The singularities of this function are at x = -1, i and -i, so the partial fraction decomposition is:

-1/2 1/(x+1) +1/4 (1+i)/(x-i) + 1/4 (1-i)/(x+i)

The integral is thus given by:

-1/2 Log(x+1) + 1/4 (1+i)Log(x-i) + cc of last term. =

-1/2 Log(x+1) + 1/2 Re[(1+i)Log(x-i)]

Real and imaginary parts of logarithms of complex arguments are easily obtained as follows. We have:

Log[r exp(i theta)] = Log(r) + i theta

This means that:

Log(x + i y) = 1/2 Log(x^2 + y^2) + i arctan(y/x)

The integral is thus given by:

-1/2 Log(x+1) + 1/4 Log(x^2 + 1) + 1/2 arctan(1/x)

-1/2 Log(x+1) + 1/4 Log(x^2 + 1) - 1/2 arctan(x)

(pi absorbed in integration constant which we don't write down)

Last edited: Feb 8, 2010
3. Feb 8, 2010

### zhentil

Um, I believe #1 is just partial fractions, while #2 is integration by parts. No need to be fancy :)

4. Feb 8, 2010

### elibj123

sin(x)ln(x)-cos(x)/x=(-cos(x))'ln(x)+(-cos(x))(ln(x))'=(-cos(x)ln(x))'

5. Feb 9, 2010

### Staff: Mentor

Welcome to the PF, Ratio. We generally do not allow homework-like brain teasers here on the PF, for obvious reasons. Please do not post this type of question again. I've moved this question to the Homework Help forums, and the normal homework rules apply (we don't do these types of questions for students).

6. Feb 12, 2010

### Ratio Test =)

One Question : Who told you this is for my homework ?