The Proof is Trivial!

Now how about

Problem 467**

$\displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{1 + x} \, dx$?

though perhaps the way I did it was over-complicated, I don't know

Solution 467

$\displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{1 + x} dx[br][br]=\displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{1} dx + \displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{x} dx$


$= \frac{\pi^2}{24}+-2+\frac{\pi}{2}+log(2)$

Using the previous result and integrating $log(1+x^2)$ by parts (basic integral, nothing worth showing).

Can you break up a fraction from the sum in the denominator like that? I thought that only worked in the numerator

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Solution 467

$\displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{1 + x} dx[br][br]=\displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{1} dx + \displaystyle \int_0^{1} \frac{\ln(1 + x^2)}{x} dx$


$= \frac{\pi^2}{24}+-2+\frac{\pi}{2}+log(2)$

Using the previous result and integrating $log(1+x^2)$ by parts (basic integral, nothing worth showing).

Not correct I'm afraid, I've never seen an integral broken up like that before but I guess you had a reason?

I don't like integrals. But you guys sure do. Here's one I can only do via complex analysis; I wonder if anyone can manage it without it:

Problem 468***

$\displaystyle\int_0^{\pi}\dfrac{\cos^2 x}{5+4\cos x}\ dx$

You're joking right?

Why would I be joking? Have you seen half the integrals in this thread.. I just wolfram'd this thing and it has one gnarly antiderivative, but even then, some integrals in this thread don't even have antiderivatives at all.

How did you get from the first line to the second line?

Can you break up a fraction from the sum in the denominator like that? I thought that only worked in the numerator

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You can't (usually) so it's either an error or a special case in this situation.

Not correct I'm afraid, I've never seen an integral broken up like that before but I guess you had a reason?

Surely you would try long division before complex analysis?? Then it's just a t-sub away

Well, I did say I don't really like integrals! If a sledge hammer I have I know will work and I have no other immediate ideas, I usually get on with it. Have you tried it, though?

Oh fair enough . Yeah I did try it and got $5\pi/24$

That's what I got. Indeed, my integration ability is embarrassing

No way, if I knew complex analysis I'd try and use it on every problem, it's pretty cool

Nah, DUTIS is way cooler. It seems like absolute magic.