Integration by Partial Fractions

In summary, the conversation is about finding the integral of (x+8)(x^2+16), with the person asking for their steps to be checked and the solution to be verified. A, B, and C are used to represent the unknown coefficients in the partial fraction decomposition, but there is an error in solving for them. The mistake is eventually realized and corrected.
  • #1
cathy
90
0

Homework Statement



1/ (x+8)(x^2+16)
Find the integral



Homework Equations



I keep getting this question wrong. Can someone check my steps?

The Attempt at a Solution



I set it up as
A/(x+8) + (Bx+C)/(x^2+16)

So I did, A(x^2+16)+ (Bx+C)(x+8)
and I did that and got
A+b=0
8B+C=0
16A+8C=1

By algebra, I solved for A, B, and C and got
A= 1/10
B= -1/10
C= 4/5

So I got 1/10ln(x+8) - 1/10∫x/(x^2+16) + 4/5∫1/(x^2+16)
and solving that out, I get 1/10ln(x+8) -1/20ln(x^2+16) + 1/5arctan(x/4)
which is not correct.
Where did I go wrong?
Thanks in advance.
 
Physics news on Phys.org
  • #2
The linear equations A, B, and C satisfy are fine, but you didn't solve for A, B, and C correctly. If you plug in your values for A and C into the last equation, you'll see it's not satisfied.
 
  • Like
Likes 1 person
  • #3
Oh I see my mistake. Such a simple error. Thank you.
 

FAQ: Integration by Partial Fractions

1. What is integration by partial fractions?

Integration by partial fractions is a mathematical technique used to integrate rational functions, which are functions that can be written as a quotient of two polynomials. It involves breaking down a complex fraction into simpler fractions that can be integrated easily.

2. When is integration by partial fractions used?

This method is typically used when integrating rational functions that cannot be integrated using other techniques, such as substitution or integration by parts. It is also useful for solving integrals involving improper fractions.

3. How do you perform integration by partial fractions?

The steps for performing integration by partial fractions are as follows:
1. Write the rational function in the form of numerator/denominator
2. Factor the denominator into irreducible factors
3. Set up the partial fraction decomposition, with each term having a constant numerator and a factor from the denominator
4. Equate the original rational function to the partial fraction decomposition and solve for the constants
5. Finally, integrate each term separately
It is important to note that the decomposition is not always unique and may require some trial and error.

4. What are some common mistakes when using integration by partial fractions?

One common mistake is forgetting to include all the terms in the partial fraction decomposition. Another mistake is incorrectly setting up the decomposition, such as using the wrong factors or not including all the necessary constants. It is also important to be careful with algebraic manipulations when solving for the constants.

5. Are there any alternatives to integration by partial fractions?

Yes, some alternatives to integration by partial fractions include using trigonometric substitutions, completing the square, or using the method of residues. However, integration by partial fractions is often the most straightforward and efficient method for integrating rational functions.

Similar threads

A doubt in Partial fraction decomposition
Replies
8
Views
1K
Partial fraction decomposition with Laplace transformation in ODE
Replies
9
Views
2K
Solve the given problem that involves integration
Replies
4
Views
1K
Expanding (i) and Solving (ii): Find the Solution
Replies
2
Views
865
Determining domain for C^1 function
Replies
4
Views
658
Laplace transform of proper rational function
Replies
6
Views
574
IVP problem using Laplace transform and partial fractions
Replies
1
Views
598
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
Replies
2
Views
867
Find f(x,y) given partial derivative and initial condition
Replies
6
Views
1K
Solve the problem that involves partial fractions
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top