Integral of x/((x-3)^2)

\int\frac{x}{\left(x-3\right)^2}dx

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Answer

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

Step by step solution

Problem

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

Solve the integral $\int\frac{x}{\left(x-3\right)^2}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=x-3 \\ du=dx\end{matrix}$
2

Rewriting $x$ in terms of $u$

$x=3+u$
3

Substituting $u$, $dx$ and $x$ in the integral

$\int\frac{3+u}{u^2}du$
4

Split the fraction $\frac{u+3}{u^2}$ in two terms with same denominator

$\int\left(\frac{3}{u^2}+\frac{u}{u^2}\right)du$
5

Simplifying the fraction by $u$

$\int\left(\frac{3}{u^2}+\frac{1}{u}\right)du$
6

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

$\int\frac{3}{u^2}du+\int\frac{1}{u}du$
7

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

$\int\frac{3}{u^2}du+\ln\left|u\right|$
8

Substitute $u$ back for it's value, $x-3$

$\int3u^{-2}du+\ln\left|x-3\right|$
9

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\int\frac{3}{u^{2}}du+\ln\left|x-3\right|$
10

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int3u^{-2}du+\ln\left|x-3\right|$
11

Taking the constant out of the integral

$3\int u^{-2}du+\ln\left|x-3\right|$
12

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$3\frac{u^{-1}}{-1}+\ln\left|x-3\right|$
13

Substitute $u$ back for it's value, $x-3$

$3\frac{\left(x-3\right)^{-1}}{-1}+\ln\left|x-3\right|$
14

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$3\frac{\frac{1}{x-3}}{-1}+\ln\left|x-3\right|$
15

Simplify the fraction

$\ln\left|x-3\right|-3\frac{1}{x-3}$
16

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-3$ and $x=x-3$

$\frac{-3}{x-3}+\ln\left|x-3\right|$
17

Add the constant of integration

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

Answer

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

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.29 seconds

Views:

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