Integral of $$$\operatorname{acos}{\left(3 x \right)}$$$

Find $$$\int \operatorname{acos}{\left(3 x \right)}\, dx$$$.

Let $$$u=3 x$$$.

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

The integral can be rewritten as

$${\color{red}{\int{\operatorname{acos}{\left(3 x \right)} d x}}} = {\color{red}{\int{\frac{\operatorname{acos}{\left(u \right)}}{3} d u}}}$$

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 $$$f{\left(u \right)} = \operatorname{acos}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\operatorname{acos}{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\int{\operatorname{acos}{\left(u \right)} d u}}{3}\right)}}$$

For the integral $$$\int{\operatorname{acos}{\left(u \right)} d u}$$$, use integration by parts $$$\int \operatorname{\omega} \operatorname{dv} = \operatorname{\omega}\operatorname{v} - \int \operatorname{v} \operatorname{d\omega}$$$.

Let $$$\operatorname{\omega}=\operatorname{acos}{\left(u \right)}$$$ and $$$\operatorname{dv}=du$$$.

Then $$$\operatorname{d\omega}=\left(\operatorname{acos}{\left(u \right)}\right)^{\prime }du=- \frac{1}{\sqrt{1 - u^{2}}} du$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d u}=u$$$ (steps can be seen »).

Thus,

$$\frac{{\color{red}{\int{\operatorname{acos}{\left(u \right)} d u}}}}{3}=\frac{{\color{red}{\left(\operatorname{acos}{\left(u \right)} \cdot u-\int{u \cdot \left(- \frac{1}{\sqrt{1 - u^{2}}}\right) d u}\right)}}}{3}=\frac{{\color{red}{\left(u \operatorname{acos}{\left(u \right)} - \int{\left(- \frac{u}{\sqrt{1 - u^{2}}}\right)d u}\right)}}}{3}$$

Let $$$v=1 - u^{2}$$$.

Then $$$dv=\left(1 - u^{2}\right)^{\prime }du = - 2 u du$$$ (steps can be seen »), and we have that $$$u du = - \frac{dv}{2}$$$.

Therefore,

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\left(- \frac{u}{\sqrt{1 - u^{2}}}\right)d u}}}}{3} = \frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\frac{1}{2 \sqrt{v}} d v}}}}{3}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(v \right)} = \frac{1}{\sqrt{v}}$$$:

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\frac{1}{2 \sqrt{v}} d v}}}}{3} = \frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\left(\frac{\int{\frac{1}{\sqrt{v}} d v}}{2}\right)}}}{3}$$

Apply the power rule $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=- \frac{1}{2}$$$:

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\frac{1}{\sqrt{v}} d v}}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{v^{- \frac{1}{2}} d v}}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\frac{v^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\left(2 v^{\frac{1}{2}}\right)}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\left(2 \sqrt{v}\right)}}}{6}$$

Recall that $$$v=1 - u^{2}$$$:

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{\sqrt{{\color{red}{v}}}}{3} = \frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{\sqrt{{\color{red}{\left(1 - u^{2}\right)}}}}{3}$$

Recall that $$$u=3 x$$$:

$$- \frac{\sqrt{1 - {\color{red}{u}}^{2}}}{3} + \frac{{\color{red}{u}} \operatorname{acos}{\left({\color{red}{u}} \right)}}{3} = - \frac{\sqrt{1 - {\color{red}{\left(3 x\right)}}^{2}}}{3} + \frac{{\color{red}{\left(3 x\right)}} \operatorname{acos}{\left({\color{red}{\left(3 x\right)}} \right)}}{3}$$

Therefore,

$$\int{\operatorname{acos}{\left(3 x \right)} d x} = x \operatorname{acos}{\left(3 x \right)} - \frac{\sqrt{1 - 9 x^{2}}}{3}$$

Add the constant of integration:

$$\int{\operatorname{acos}{\left(3 x \right)} d x} = x \operatorname{acos}{\left(3 x \right)} - \frac{\sqrt{1 - 9 x^{2}}}{3}+C$$

$$$\int \operatorname{acos}{\left(3 x \right)}\, dx = \left(x \operatorname{acos}{\left(3 x \right)} - \frac{\sqrt{1 - 9 x^{2}}}{3}\right) + C$$$A