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

[nerdy_hottie]

Homework Statement

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 !

Homework 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.

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.

 

[Mark44]

Homework Statement

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 !

Homework 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.

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.

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.

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.

I don't have any ideas just yet, but I'll think about this one.

3. I have no ideas on this one.

If u = cos^-1(x), what is du?

BTW, welcome to Physics Forums!

 

[SteamKing]

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.

 

[SteamKing]

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.

Check that last suggestion.

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

 

[nerdy_hottie]

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.

 

[vela]

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=e^u 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.