Math Help - Help with Integral Problems

1. Help with Integral Problems

I have three in particular, two of which I can't seem to get anywhere and another which I think I may have domain issues.

-Just for reference, I can only solve these using u-substitution or through anti-derivatives that we've memorized. (i.e. sinx'=cosx, x^n'=n-1x^n-1, etc.)

1.) $\int^\frac{\pi}{4}_\frac{-\pi}{4} \frac{t^4tan(t)}{2+cos(t)}\,dx$

I've tried u-substitution for this one but when finding du/dx nothing seems to replace every t variable to simplify into something that's easy to integrate. I've tried rewriting but rewriting it makes it even more complicated. (I've tried rewriting tan(t) into sin(t)/cos(t), I've tried to split up the problem, but no avail, nothing seems to work) A few tips on how to start would greatly help!!

2.) $\int^1_0 \frac{e^x}{1+e^{2x}}\,dx$

On this one, I've tried u-substitution with the denominator & numerator portion of the function:

u=1+e^2x
du/dx=e^2x * 2
1/2du=e^2x dx

or

u=e^x
du=e^x dx

Neither one replaces every x variable in the original function so I'm stuck there.

3.) $\int^{10}_{1} \frac{x}{x^2-4}\,dx$

This last one I was able to integrate but once I start evaluating for the x intervals it asks for I get a domain issue. After integrating and using u-substitution I get

F(x) = 1/2 ln (x^2 - 4) from x-values 10 and 1.

When plugging in 10 I get a positive value which is solvable; however, when plugging in 1 I get a negative value which isn't defined at ln(x).

Nevermind, solved this one

-All the help and assistance will be greatly appreciated!!!

2. Re: Help with Integral Problems

second one

$\int _{0}^{1} \frac{ e^x}{1+ e^{2x}} \; dx$

let $u = e^x \Rightarrow du = e^x dx \Rightarrow \frac{du}{u} = dx$

$\int \frac{u}{1+u^2} \left(\frac{du}{u}\right)$

$\int \frac{u}{u(1+u^2)} \; du$

$\int \frac{1}{1+u^2} \; du = \tan ^{-1} (u) + C$

3. Re: Help with Integral Problems

The first one cannot be solved in terms of the standard functions. The third one is undefined.

4. Re: Help with Integral Problems

Originally Posted by princeps
The first one cannot be solved in terms of the standard functions. The third one is undefined.
Thank you, I kinda of find it odd that our would teacher would assign that problem considering we haven't covered the material necessary to solve it, but then again, we had a similar issue with a different problem that could not be solved with standard functions, and she let us omit it . I'll just bring it to her attention tomorrow in class.

5. Re: Help with Integral Problems

Originally Posted by dagbayani481
I have three in particular, two of which I can't seem to get anywhere and another which I think I may have domain issues.

-Just for reference, I can only solve these using u-substitution or through anti-derivatives that we've memorized. (i.e. sinx'=cosx, x^n'=n-1x^n-1, etc.)

1.) $\int^\frac{\pi}{4}_\frac{-\pi}{4} \frac{t^4tan(t)}{2+cos(t)}\,dx$

I've tried u-substitution for this one but when finding du/dx nothing seems to replace every t variable to simplify into something that's easy to integrate. I've tried rewriting but rewriting it makes it even more complicated. (I've tried rewriting tan(t) into sin(t)/cos(t), I've tried to split up the problem, but no avail, nothing seems to work) A few tips on how to start would greatly help!!
The integrand is an odd function, and you are integrating over a symmetric interval. Can you infer anything from that?