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Exercise

$\int\ln\sqrt{x^2-9}dx$

Step-by-step Solution

1

Apply properties of logarithms to expand and simplify the logarithmic expression $\ln\left(\sqrt{x^2-9}\right)$ inside the integral

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

Expand the integral $\int\left(\frac{1}{2}\ln\left(x+3\right)+\frac{1}{2}\ln\left(x-3\right)\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

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

The integral $\int\frac{1}{2}\ln\left(x+3\right)dx$ results in: $\frac{1}{2}\left(\left(x+3\right)\ln\left(x+3\right)-x-3\right)$

$\frac{1}{2}\left(\left(x+3\right)\ln\left(x+3\right)-x-3\right)$
4

The integral $\int\frac{1}{2}\ln\left(x-3\right)dx$ results in: $\frac{1}{2}\left(\left(x-3\right)\ln\left(x-3\right)-x+3\right)$

$\frac{1}{2}\left(\left(x-3\right)\ln\left(x-3\right)-x+3\right)$
5

Gather the results of all integrals

$\frac{1}{2}\left(\left(x+3\right)\ln\left|x+3\right|-x-3\right)+\frac{1}{2}\left(\left(x-3\right)\ln\left|x-3\right|-x+3\right)$
6

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

$\frac{1}{2}\left(\left(x+3\right)\ln\left|x+3\right|-x-3\right)+\frac{1}{2}\left(\left(x-3\right)\ln\left|x-3\right|-x+3\right)+C_0$

Final answer to the exercise

$\frac{1}{2}\left(\left(x+3\right)\ln\left|x+3\right|-x-3\right)+\frac{1}{2}\left(\left(x-3\right)\ln\left|x-3\right|-x+3\right)+C_0$

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  • Integrate using basic integrals
  • Product of Binomials with Common Term
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Main Topic: Integrals involving Logarithmic Functions

They are those integrals where the function that we are integrating is composed only of combinations of logarithmic functions.

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