# I'm having trouble finding the integral using u-substitution.

1. ### nerdy_hottie

19
1. The problem statement, all variables and given/known data

I have to integrate using u-substitution (probably).

Ex. 1 Integrate (sin^4x)/(cos^6x)dx

2. Integrate (2x)/(sqrt(e^(2x^2)-1))dx

3. Integrate (cos^-1x)/(sqrt(1-x^2))dx

Thank you !

2. Relevant equations

I do not want the solutions. I just need to be pointed in the right direction (i.e. I need you to help me start off)

**It should be noted that I am doing a calculus II course (Integral Calc, mostly) in university, so it's not very advanced integrals that I'm doing. Basically what I know is how to integrate using u-substitution, and I know the integrals for the inverse trig functions (which is supposed to be relevant to examples 2 & 3), and that's what information I have to work with.

**It should also be noted that I may just not know how to rewrite the equations before I can integrate them. I have trouble 'seeing through' the equation and automatically knowing which way I'm going to solve it.

3. The attempt at a solution

Ex. 1 I tried rewriting the equation using trig identities, e.g. (1-cos(x))/(1-sin(x))^3. I found this got me nowhere.
I also tried rewriting it is (sin^4)(x)/(cos^4)(x)*1/cos(x), and rewriting and rewriting until I ended up with a big mess, so that got me nowhere as well.

2. Here's my dilemma:
-if I substitute e^2x for u, I end up needing an e to the power in my numerator, so that doesn't work out.
-if I instead substitute 2x^2 for u, I end up with the e to the power of u on the bottom and I don't have a formula for that.

3. I have no ideas on this one.

### Staff: Mentor

This is the right approach, but you have an error. Your integrand is equal to tan4(x)sec2(x). That should suggest a pretty obvious substitution.
If u = cos-1(x), what is du?

BTW, welcome to Physics Forums!

3. ### SteamKing

10,942
Staff Emeritus
For example 2, rewrite the integrand to remove the sqrt in the denominator, that is, express 1/sqrt(e^(2*x^2)-1) using the appropriate exponent. After doing this, see if the factor 2x would be useful in integration by parts.

4. ### SteamKing

10,942
Staff Emeritus
Check that last suggestion.

See if the factor 2x would be useful in a u-substitution integration.

5. ### nerdy_hottie

19
Thanks all, but I still cannot find the solution to the second example.

I let u=2x, so du=2dx
Since the x is is still in the numerator, I say that also, x=u/2
So I fill this in and I get Integral of (u/2)(1/(sqrt((e^u)-1))du

I have not yet learned to do integration by parts, by the way.

6. ### vela

12,963
Staff Emeritus
The argument of the exponential has an x2 in it, right? So try u=x2 to try simplify that a bit. That's where you find the factor of 2x comes in handy.

Then you might try a substitution like v=eu and see where that gets you. A lot of this you figure out by trial and error. As you do more problems, you'll start to get a feel for what works and what doesn't.