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\int\frac{5x-2}{x^2+2x-8}dx

Integral of (5x-2)/(x^2+2x-8)

Answer

$\frac{11}{3}\ln\left|x+4\right|+\frac{4}{3}\ln\left|x-2\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int\frac{5x-2}{x^2+2x-8}dx$
1

We can factor the polynomial $x^2+2x-8$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $-8$ and we found that $2$ is a root of the polynomial

$2^2+2\cdot 2-8=0$
2

Let's divide the polynomial by $x-2$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $2$. Add the result to the second coefficient and then multiply this by $2$ and so on

$\left|\begin{array}{c}1 & 2 & -8 \\ & 2 & 8 \\ 1 & 4 & 0\end{array}\right|2$

Unlock this step-by-step solution!

Answer

$\frac{11}{3}\ln\left|x+4\right|+\frac{4}{3}\ln\left|x-2\right|+C_0$
$\int\frac{5x-2}{x^2+2x-8}dx$

Main topic:

Integration by substitution

Used formulas:

1. See formulas

Time to solve it:

~ 1.43 seconds

Related topics:


Integration by substitution