# Solve for: integral of (4x)/(x^2-8x+16) x

## Expression: $\int{ \frac{ 4x }{ {x}^{2}-8x+16 } } \mathrm{d} x$

Use ${a}^{2}-2ab+{b}^{2}={\left( a-b \right)}^{2}$ to factor the expression

$\int{ \frac{ 4x }{ {\left( x-4 \right)}^{2} } } \mathrm{d} x$

Use the substitution $t=x-4$ to transform the integral

$\int{ \frac{ 4t+16 }{ {t}^{2} } } \mathrm{d} t$

Separate the fraction into $2$ fractions

$\int{ \frac{ 4t }{ {t}^{2} }+\frac{ 16 }{ {t}^{2} } } \mathrm{d} t$

Cancel out the common factor $t$

$\int{ \frac{ 4 }{ t }+\frac{ 16 }{ {t}^{2} } } \mathrm{d} t$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\int{ \frac{ 4 }{ t } } \mathrm{d} t+\int{ \frac{ 16 }{ {t}^{2} } } \mathrm{d} t$

Use $\int{ \frac{ a }{ x } } \mathrm{d} x=a \times \ln\left({|x|}\right)$ to evaluate the integral

$4\ln\left({|t|}\right)+\int{ \frac{ 16 }{ {t}^{2} } } \mathrm{d} t$

Evaluate the indefinite integral

$4\ln\left({|t|}\right)-\frac{ 16 }{ t }$

Substitute back $t=x-4$

$4\ln\left({|x-4|}\right)-\frac{ 16 }{ x-4 }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }4\ln\left({|x-4|}\right)-\frac{ 16 }{ x-4 }+C,& C \in ℝ\end{array}$

Random Posts
Random Articles