Integral of $$$\frac{1}{5 - 3 x^{2}}$$$

Find $$$\int \frac{1}{5 - 3 x^{2}}\, dx$$$.

Perform partial fraction decomposition (steps can be seen »):

$${\color{red}{\int{\frac{1}{5 - 3 x^{2}} d x}}} = {\color{red}{\int{\left(\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} - \frac{\sqrt{15}}{10 \left(3 x - \sqrt{15}\right)}\right)d x}}}$$

Integrate term by term:

$${\color{red}{\int{\left(\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} - \frac{\sqrt{15}}{10 \left(3 x - \sqrt{15}\right)}\right)d x}}} = {\color{red}{\left(- \int{\frac{\sqrt{15}}{10 \left(3 x - \sqrt{15}\right)} d x} + \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{\sqrt{15}}{10}$$$ and $$$f{\left(x \right)} = \frac{1}{3 x - \sqrt{15}}$$$:

$$\int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - {\color{red}{\int{\frac{\sqrt{15}}{10 \left(3 x - \sqrt{15}\right)} d x}}} = \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - {\color{red}{\left(\frac{\sqrt{15} \int{\frac{1}{3 x - \sqrt{15}} d x}}{10}\right)}}$$

Let $$$u=3 x - \sqrt{15}$$$.

Then $$$du=\left(3 x - \sqrt{15}\right)^{\prime }dx = 3 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{3}$$$.

Thus,

$$\int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - \frac{\sqrt{15} {\color{red}{\int{\frac{1}{3 x - \sqrt{15}} d x}}}}{10} = \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - \frac{\sqrt{15} {\color{red}{\int{\frac{1}{3 u} d u}}}}{10}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$\int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - \frac{\sqrt{15} {\color{red}{\int{\frac{1}{3 u} d u}}}}{10} = \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - \frac{\sqrt{15} {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{3}\right)}}}{10}$$

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$\int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - \frac{\sqrt{15} {\color{red}{\int{\frac{1}{u} d u}}}}{30} = \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} - \frac{\sqrt{15} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{30}$$

Recall that $$$u=3 x - \sqrt{15}$$$:

$$- \frac{\sqrt{15} \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{30} + \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x} = - \frac{\sqrt{15} \ln{\left(\left|{{\color{red}{\left(3 x - \sqrt{15}\right)}}}\right| \right)}}{30} + \int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{\sqrt{15}}{10}$$$ and $$$f{\left(x \right)} = \frac{1}{3 x + \sqrt{15}}$$$:

$$- \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + {\color{red}{\int{\frac{\sqrt{15}}{10 \left(3 x + \sqrt{15}\right)} d x}}} = - \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + {\color{red}{\left(\frac{\sqrt{15} \int{\frac{1}{3 x + \sqrt{15}} d x}}{10}\right)}}$$

Let $$$u=3 x + \sqrt{15}$$$.

Then $$$du=\left(3 x + \sqrt{15}\right)^{\prime }dx = 3 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{3}$$$.

So,

$$- \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} {\color{red}{\int{\frac{1}{3 x + \sqrt{15}} d x}}}}{10} = - \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} {\color{red}{\int{\frac{1}{3 u} d u}}}}{10}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$- \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} {\color{red}{\int{\frac{1}{3 u} d u}}}}{10} = - \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{3}\right)}}}{10}$$

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$- \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} {\color{red}{\int{\frac{1}{u} d u}}}}{30} = - \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{30}$$

Recall that $$$u=3 x + \sqrt{15}$$$:

$$- \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{30} = - \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} \ln{\left(\left|{{\color{red}{\left(3 x + \sqrt{15}\right)}}}\right| \right)}}{30}$$

Therefore,

$$\int{\frac{1}{5 - 3 x^{2}} d x} = - \frac{\sqrt{15} \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)}}{30} + \frac{\sqrt{15} \ln{\left(\left|{3 x + \sqrt{15}}\right| \right)}}{30}$$

Simplify:

$$\int{\frac{1}{5 - 3 x^{2}} d x} = \frac{\sqrt{15} \left(- \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)} + \ln{\left(\left|{3 x + \sqrt{15}}\right| \right)}\right)}{30}$$

Add the constant of integration:

$$\int{\frac{1}{5 - 3 x^{2}} d x} = \frac{\sqrt{15} \left(- \ln{\left(\left|{3 x - \sqrt{15}}\right| \right)} + \ln{\left(\left|{3 x + \sqrt{15}}\right| \right)}\right)}{30}+C$$

$$$\int \frac{1}{5 - 3 x^{2}}\, dx = \frac{\sqrt{15} \left(- \ln\left(\left|{3 x - \sqrt{15}}\right|\right) + \ln\left(\left|{3 x + \sqrt{15}}\right|\right)\right)}{30} + C$$$A