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

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In summary: Just keep practicing!In summary, the student is struggling to integrate using u-substitution and is seeking help in starting off with the problem. They have attempted to use trig identities and have tried rewriting the equations but have not been successful. They are looking for direction and have stated that they have not yet learned integration by parts. They have received suggestions to try substitutions such as u=x^2 and v=eu, and to use the factor of 2x in integration by parts.
  • #1
nerdy_hottie
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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.
 
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  • #2
nerdy_hottie said:

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.
nerdy_hottie said:
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.
nerdy_hottie said:
3. I have no ideas on this one.
If u = cos-1(x), what is du?

BTW, welcome to Physics Forums!
 
  • #3
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 said:
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.
 
  • #5
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
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.
 

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

1. What is u-substitution?

U-substitution is a technique used in calculus to solve integrals by substituting a variable, usually represented as u, for a more complicated expression within the integrand. This allows for easier integration and can help solve integrals that would otherwise be difficult or impossible to solve.

2. How do I know when to use u-substitution?

U-substitution is typically used when the integrand contains a function and its derivative, as well as when the integrand contains a polynomial expression within a radical. It can also be used to simplify integrals with trigonometric functions or exponential functions.

3. What are the steps for using u-substitution?

The steps for using u-substitution are as follows:
1. Identify the variable or expression within the integrand that can be substituted with u.
2. Differentiate u to find du.
3. Substitute the expression for u and du into the integral.
4. Simplify the integral and solve for the original variable.
5. Substitute the original variable back into the solution.

4. Can u-substitution be used for all integrals?

No, u-substitution is not always applicable to all integrals. It is most effective for integrals with specific types of expressions, such as those involving functions and their derivatives or polynomial expressions within radicals. There are other integration techniques, such as integration by parts or partial fractions, that may be more suitable for different types of integrals.

5. What should I do if I'm having trouble finding the integral using u-substitution?

If you are having trouble finding the integral using u-substitution, it may be helpful to try a different integration technique or to review the problem and make sure you have correctly identified the variable to substitute for u. You can also consult with a tutor or your instructor for further assistance.

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