- #1

csi86

- 7

- 0

1.[tex] \int \frac{x}{\sqrt{x^{2}+2x+2}} \; dx [/tex]

2.[tex] \int \frac{x}{x^{3}+1} \; dx [/tex]

Thank you for your time.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter csi86
- Start date

In summary, the conversation discussed two integrals that the person was struggling with and requested hints on how to solve them. The respondent provided hints and suggestions on how to approach the problems, such as using integration by parts and partial fractions. The conversation concluded with the person successfully solving the second integral and figuring out the correct solution for the first one.

- #1

csi86

- 7

- 0

1.[tex] \int \frac{x}{\sqrt{x^{2}+2x+2}} \; dx [/tex]

2.[tex] \int \frac{x}{x^{3}+1} \; dx [/tex]

Thank you for your time.

Physics news on Phys.org

- #2

siddharth

Homework Helper

Gold Member

- 1,143

- 0

You need to show some work before you receive help. What are your thoughts/ideas on these problem? What have you done till now, and where are you stuck?

- #3

- 13,360

- 3,470

[tex] \frac{x}{x^{3}+1} =\frac{A}{x+1} +\frac{Bx+C}{x^{2}-x+1} [/tex]

Daniel.

- #4

courtrigrad

- 1,236

- 2

For the first one use integration by parts:

[tex] u = x [/tex][tex] dv = \frac{1}{\sqrt{1+(x+1)^{2}}}\; dx[/tex]

[tex] u = x [/tex][tex] dv = \frac{1}{\sqrt{1+(x+1)^{2}}}\; dx[/tex]

Last edited:

- #5

csi86

- 7

- 0

[tex] -ln|x+1| + \frac{1}{2} ln{(x^{2}-x+1)} + \frac{2}{\sqrt{3}}\arctan{\frac{2x-1}{\sqrt{3}}} [/tex]

- #6

- 13,360

- 3,470

Just don't forget the integration constant.

Daniel.

Daniel.

- #7

csi86

- 7

- 0

I tried to solve the first one and I end up with :

[tex] \frac{x^{2}}{2 \sqrt{x^{2}+2x+2}} + \frac{1}{2} \int \frac{(x^{2})(2x+2)}{2 \sqrt{(x^{2}+2x+2})^{3}}} \;dx [/tex]

(Using integration by parts)

- #8

- 13,360

- 3,470

[tex] \int \frac{x}{\sqrt{x^{2}+2x+2}} \ dx =\frac{1}{2}\int \frac{d(x^{2}+2x+2)}{\sqrt{x^{2}+2x+2}} -\int \frac{dx}{\sqrt{x^{2}+2x+2}} [/tex]

Daniel.

- #9

csi86

- 7

- 0

I end up with :

[tex] \sqrt{x^{2}+2x+2} - \ln{(x+1+ \sqrt{x^{2}+2x+2})} [/tex]

I don't know if that is correct but I honestly hope so :).

Antiderivatives are the inverse operation of derivatives. They are important for exams because they allow us to find the original function from its derivative, which is often necessary to solve problems in calculus.

Antiderivatives can be tricky because there is no one set rule for finding them. Instead, you must use a combination of techniques such as substitution, integration by parts, and trigonometric identities. To master antiderivatives, it is important to practice and familiarize yourself with these techniques.

One helpful hint is to look for patterns in the function and try to use known antiderivative formulas. Another hint is to try different techniques if one approach is not working. Additionally, make sure to always double check your work and simplify your final answer.

The key to improving speed is practice. The more you practice solving antiderivatives, the more familiar you will become with the different techniques and the quicker you will be able to recognize which technique to use for a given problem. It is also helpful to work on your algebra skills to simplify your calculations.

One common mistake is forgetting to add the constant of integration when finding the antiderivative. Another mistake is not simplifying the final answer, which can lead to incorrect solutions. It is also important to pay attention to the limits of integration and adjust accordingly when solving definite integrals.

- Replies
- 22

- Views
- 2K

- Replies
- 3

- Views
- 826

- Replies
- 15

- Views
- 1K

- Replies
- 5

- Views
- 2K

- Replies
- 8

- Views
- 1K

- Replies
- 10

- Views
- 1K

- Replies
- 7

- Views
- 1K

- Replies
- 19

- Views
- 2K

- Replies
- 12

- Views
- 2K

- Replies
- 2

- Views
- 1K

Share: