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

La calculadora encontrará la integral/antiderivada de $\operatorname{acos}{\left(x \right)}$, con los pasos que se muestran.

Escriba sin diferenciales como $dx$, $dy$ etc.
Deje vacío para la detección automática.

Si la calculadora no calculó algo o ha identificado un error, o tiene una sugerencia/comentario, escríbalo en los comentarios a continuación.

Tu aportación

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

Solución

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

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

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

The integral becomes

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

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

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

So,

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

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

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

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

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

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

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

Therefore,

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

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