Solve the integral of logarithmic functions int(ln(4x^2+8x+4))dx

 Exercise

$\int ln\left(4x^2+\:8\:x\:+\:4\right)dx$

 

Apply properties of logarithms to expand and simplify the logarithmic expression $\ln\left(4x^2+8x+4\right)$ inside the integral

$\int2\ln\left(2x+2\right)dx$

 

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

$2\int\ln\left(2x+2\right)dx$

 

The integral $\int\ln\left(2x+2\right)dx$ results in $\left(2x+2\right)\ln\left(2x+2\right)-\left(2x+2\right)$

$2\left(\left(2x+2\right)\ln\left|2x+2\right|-\left(2x+2\right)\right)$

 

Simplify the product $-(2x+2)$

$2\left(\left(2x+2\right)\ln\left|2x+2\right|-2x-2\right)$

 

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$2\left(\left(2x+2\right)\ln\left|2x+2\right|-2x-2\right)+C_0$

$2\left(\left(2x+2\right)\ln\left|2x+2\right|-2x-2\right)+C_0$

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