# Integrals by partial fraction expansion Calculator

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### Difficult Problems

1

Example

$\int\frac{2x-1}{x^2-8x+15}dx$
2

Factor the trinomial $15-8x+x^2$ finding two numbers that multiply to form $15$ and added form $-8$

$\begin{matrix}\left(-5\right)\left(-3\right)=15\\ \left(-5\right)+\left(-3\right)=-8\end{matrix}$
3

Thus

$\int\frac{2x-1}{\left(x-3\right)\left(x-5\right)}dx$
4

Using partial fraction decomposition, the fraction $\frac{2x-1}{\left(x-3\right)\left(x-5\right)}$ can be rewritten as

$\frac{2x-1}{\left(x-3\right)\left(x-5\right)}=\frac{A}{x-3}+\frac{B}{x-5}$
5

Now we need to find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x-3\right)\left(x-5\right)$

$2x-1=\left(\frac{A}{x-3}+\frac{B}{x-5}\right)\left(x-3\right)\left(x-5\right)$
6

Multiplying polynomials

$2x-1=\frac{A\left(x-3\right)\left(x-5\right)}{x-3}+\frac{B\left(x-3\right)\left(x-5\right)}{x-5}$
7

Simplifying

$2x-1=A\left(x-5\right)+B\left(x-3\right)$
8

Expand the polynomial

$2x-1=A\left(x-5\right)+B\left(x-3\right)$
9

Assigning values to $x$ we obtain the following system of equations

$\begin{matrix}-7=-6B-8A&\:\:\:\:\:\:\:(x=-3) \\ 5=-2A&\:\:\:\:\:\:\:(x=3)\end{matrix}$
10

Proceed to solve the system of linear equations

$\begin{matrix} -8A & - & 6B & =-7 \\ -2A & + & 0B & =5\end{matrix}$
11

Rewrite as a coefficient matrix

$\left(\begin{matrix}-8 & -6 & -7 \\ -2 & 0 & 5\end{matrix}\right)$
12

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & -\frac{5}{2} \\ 0 & 1 & \frac{9}{2}\end{matrix}\right)$
13

The decomposed integral equivalent is

$\int\left(\frac{-\frac{5}{2}}{x-3}+\frac{\frac{9}{2}}{x-5}\right)dx$
14

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int\frac{-\frac{5}{2}}{x-3}dx+\int\frac{\frac{9}{2}}{x-5}dx$
15

Apply the formula: $\int\frac{n}{b+x}dx$$=n\ln\left|b+x\right|, where b=-3 and n=-\frac{5}{2} \int\frac{\frac{9}{2}}{x-5}dx-\frac{5}{2}\ln\left|x-3\right| 16 Apply the formula: \int\frac{n}{b+x}dx$$=n\ln\left|b+x\right|$, where $b=-5$ and $n=\frac{9}{2}$

$\frac{9}{2}\ln\left|x-5\right|-\frac{5}{2}\ln\left|x-3\right|$
17

$-\frac{5}{2}\ln\left|x-3\right|+\frac{9}{2}\ln\left|x-5\right|+C_0$