How Can I Solve These Complex Integrals?

  • Context: Undergrad 
  • Thread starter Thread starter Alexx1
  • Start date Start date
  • Tags Tags
    Dx
Click For Summary

Discussion Overview

The discussion revolves around solving three specific integrals, exploring various techniques and substitutions for integration. Participants share their approaches, corrections, and challenges encountered while attempting to solve these integrals.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Participants present three integrals for discussion: 1/(1+sqrt(x)) dx, (x^3)*(e^x^2), and (x*e^x)/((x+1)^2).
  • Some participants suggest using substitutions for the first two integrals, such as letting 1+sqrt(x)=u and x^2=u.
  • Integration by parts is proposed for the second and third integrals, with specific variable assignments for u and v'.
  • One participant expresses confusion regarding the correct substitution and differentiation process, particularly for the first integral.
  • Another participant points out a mistake in the factor of x when substituting u=x^2, clarifying the correct relationship between dx and du.
  • There is a discussion about the constant of integration and how it relates to the results obtained from the integrals.
  • One participant successfully solves one of the integrals but seeks further assistance with another integral involving trigonometric functions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best methods for solving the integrals, and there are multiple competing views regarding the correct substitutions and integration techniques. Some participants correct or challenge each other's approaches without resolving the disagreements.

Contextual Notes

Some participants express uncertainty about the steps involved in their substitutions and the implications of their integration results, indicating that further clarification is needed on specific mathematical processes.

Who May Find This Useful

Students and individuals interested in calculus, particularly those seeking assistance with integration techniques and problem-solving strategies for complex integrals.

Alexx1
Messages
86
Reaction score
0
How do you solve these 3 integrals? :

Integral 1 : 1/(1+sqrt(x)) dx

Integral 2: (x^3)*(e^x^2)

Integral 3: (x*e^x)/((x+1)^2)

I have no idea how to solve these integrals..
 
Physics news on Phys.org
Hi Alexx1! :smile:

(have an integral: ∫ and a square-root: √ and try using the X2 tag just above the Reply box :wink:)
Alexx1 said:
How do you solve these 3 integrals? :

Integral 1 : 1/(1+sqrt(x)) dx

Integral 2: (x^3)*(e^x^2)

Integral 3: (x*e^x)/((x+1)^2)

I have no idea how to solve these integrals..

For 1 and 2, use the obvious substitutions. :wink:
 


1) let 1+\sqrt{x}=u

2) let x^2=u and then apply the integration by parts.

3) Apply integration by parts by letting u=xe^x and v'=(x+1)^{-2}


If you're still stuck after this, show us what you've done and we'll help you further. Good luck!
 


Mentallic said:
1) let 1+\sqrt{x}=u

2) let x^2=u and then apply the integration by parts.

3) Apply integration by parts by letting u=xe^x and v'=(x+1)^{-2}


If you're still stuck after this, show us what you've done and we'll help you further. Good luck!

Thx! I found the third one.
But the first and the second one, is it like

u = 1+sqrt(x) --> du = 1/2sqrt(x) dx --> dx = 2sqrt(x) du
u = x^2 --> du = 2xdx --> dx = du/2


?

(I learned to use t = ... --> dt = ... dx)
 
Last edited by a moderator:


Alexx1 said:
Thx! I found the third one.
But the first and the second one, is it like

u = 1+sqrt(x) --> du = 1/2sqrt(x) dx --> dx = 2sqrt(x) du
and sqrt(x)= u- 1 so dx= 2(u- 1)du

u = x^2 --> du = 2xdx --> dx = du/2
Yes, that's right.

?

(I learned to use t = ... --> dt = ... dx)
 


HallsofIvy said:
and sqrt(x)= u- 1 so dx= 2(u- 1)du


Yes, that's right.

For the first one I become:

integral: 2(u-1)/u du

= 2 (integral u/u du - integral 1/u du)

= 2 (u - ln u)

= 2 (sqrt(x)+1 - ln (sqrt(x)+1))

But the correct answer is: 2 (sqrt(x) - ln (sqrt(x)+1))

What have I done wrong?
 


HallsofIvy said:
Alexx1 said:
u = x^2 --> du = 2xdx --> dx = du/2
Yes, that's right.
No, that's wrong. Alex dropped a factor of x.

The substitution u=x^2 does lead to du=2x\,dx. Solving for dx, dx=1/(2x)\,du, not du/2.

Alexx1: Try using this mixed form. (Alternately, look for a 2x\,dx in the integral.) using dx=1/(2\sqrt u)\,du will just lead to confusion.
 


HallsofIvy said:
and sqrt(x)= u- 1 so dx= 2(u- 1)du


Yes, that's right.

D H said:
No, that's wrong. Alex dropped a factor of x.

The substitution u=x^2 does lead to du=2x\,dx. Solving for dx, dx=1/(2x)\,du, not du/2.

Alexx1: Try using this mixed form. (Alternately, look for a 2x\,dx in the integral.) using dx=1/(2\sqrt u)\,du will just lead to confusion.

I don't know how to find a 2x dx in the integral..
Can you explain it to me?
 


Alexx1 said:
= 2 (sqrt(x)+1 - ln (sqrt(x)+1))

But the correct answer is: 2 (sqrt(x) - ln (sqrt(x)+1))

What have I done wrong?

What you have done wrong is that you failed to realize that the extra 2 in the answer you got can be attached to the constant of integration.
Take the derivative of both and you'll have the same result :wink:
 
  • #10


Alexx1 said:
I don't know how to find a 2x dx in the integral..
Can you explain it to me?
The integral in question is

\int x^3\,e^{x^2}\,dx

Rewrite this as

\int x^2\,e^{x^2}\,xdx
 
  • #11


Mentallic said:
What you have done wrong is that you failed to realize that the extra 2 in the answer you got can be attached to the constant of integration.
Take the derivative of both and you'll have the same result :wink:

Thanks!
 
  • #12


D H said:
The integral in question is

\int x^3\,e^{x^2}\,dx

Rewrite this as

\int x^2\,e^{x^2}\,xdx

If du = 2xdx than xdx = du/2 ..

Than you get: (1/2) * integral x^2 e^u du..

Or am I wrong?
 
  • #13


Alexx1 said:
If du = 2xdx than xdx = du/2 ..

Than you get: (1/2) * integral x^2 e^u du..

Or am I wrong?
Correct -- but incomplete. Why did you make the u-substitution in the exponential but not for the rest of integral?
 
  • #14


D H said:
Correct -- but incomplete. Why did you make the u-substitution in the exponential but not for the rest of integral?


Ow you're right.. Stupid mistake.. Thank you very much, now I've solved it!

Can you check my other post please?

https://www.physicsforums.com/showthread.php?t=369488
 
  • #15


D H said:
Correct -- but incomplete. Why did you make the u-substitution in the exponential but not for the rest of integral?

Can you also help me with this integral?

1/(1+cos(x)+sin(x)) dx
 
  • #16


Mentallic said:
What you have done wrong is that you failed to realize that the extra 2 in the answer you got can be attached to the constant of integration.
Take the derivative of both and you'll have the same result :wink:

Can you also help me with this integral?

1/(1+cos(x)+sin(x)) dx
 

Similar threads

High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
3K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
High School Arc Length for Hyperbolic Sin
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Integration with different infinitesimal intervals
  • · Replies 31 ·
2
Replies
31
Views
4K
High School Is this identity containing the Gaussian Integral of any use?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Substitution in a definite integral
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Derive the orthonormality condition for Legendre polynomials
  • · Replies 30 ·
2
Replies
30
Views
2K
Undergrad Solving the Difficult Integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##
  • · Replies 5 ·
Replies
5
Views
4K