Integral of (2x+5)/(5x-7+x^2)

\int\frac{2x+5}{x^2+5x-7}dx

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Answer

$\ln\left|-7+5x+x^2\right|+C_0$

Step by step solution

Problem

$\int\frac{2x+5}{x^2+5x-7}dx$
1

Solve the integral $\int\frac{5+2x}{-7+5x+x^2}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=-7+5x+x^2 \\ du=\left(5+2x\right)dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{\left(5+2x\right)}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{1}{u}du$
4

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\ln\left|u\right|$
5

Substitute $u$ back for it's value, $-7+5x+x^2$

$\ln\left|-7+5x+x^2\right|$
6

Add the constant of integration

$\ln\left|-7+5x+x^2\right|+C_0$

Answer

$\ln\left|-7+5x+x^2\right|+C_0$

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Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.35 seconds

Views:

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