Can You Solve These Advanced Calculus Integrals?

[Ratio Test =)]

Hello :)
Here we go! :

1. <br /> \int \frac{dx}{x^3+x^2+x+1}<br />

2. <br /> \int_{\frac{\pi}{2}}^{\pi} \left( sin(x) ln(x) - \frac{cos(x)}{x}\right) dx<br />

Do your best :)

 

[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)

 

[zhentil]

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

 

[elibj123]

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

 

[berkeman]

Hello :)
Here we go! :

1. <br /> \int \frac{dx}{x^3+x^2+x+1}<br />

2. <br /> \int_{\frac{\pi}{2}}^{\pi} \left( sin(x) ln(x) - \frac{cos(x)}{x}\right) dx<br />

Do your best :)

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).

 

[Ratio Test =)]

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).

One Question : Who told you this is for my homework ?
Did you read the thread's title ?

 

[berkeman]

One Question : Who told you this is for my homework ?
Did you read the thread's title ?

Don't imply I'm an idiot. If you don't understand why homework questions aren't allowed as brain teasers, please think about it a bit more. And re-read the PF Rules link at the top of the page that you agreed to when you joined here.

Thread locked. Show some brains people.