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

The calculator will find the integral/antiderivative of acos(x)\operatorname{acos}{\left(x \right)}, with steps shown.

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

Solution

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

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

Then du=(acos(x))dx=11x2dx\operatorname{du}=\left(\operatorname{acos}{\left(x \right)}\right)^{\prime }dx=- \frac{1}{\sqrt{1 - x^{2}}} dx (steps can be seen ») and v=1dx=x\operatorname{v}=\int{1 d x}=x (steps can be seen »).

So,

acos(x)dx=(acos(x)xx(11x2)dx)=(xacos(x)(x1x2)dx){\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=1x2u=1 - x^{2}.

Then du=(1x2)dx=2xdxdu=\left(1 - x^{2}\right)^{\prime }dx = - 2 x dx (steps can be seen »), and we have that xdx=du2x dx = - \frac{du}{2}.

The integral can be rewritten as

xacos(x)(x1x2)dx=xacos(x)12udux \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 cf(u)du=cf(u)du\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du with c=12c=\frac{1}{2} and f(u)=1uf{\left(u \right)} = \frac{1}{\sqrt{u}}:

xacos(x)12udu=xacos(x)(1udu2)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 undu=un+1n+1\int u^{n}\, du = \frac{u^{n + 1}}{n + 1} (n1)\left(n \neq -1 \right) with n=12n=- \frac{1}{2}:

xacos(x)1udu2=xacos(x)u12du2=xacos(x)u12+112+12=xacos(x)(2u12)2=xacos(x)(2u)2x \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=1x2u=1 - x^{2}:

xacos(x)u=xacos(x)(1x2)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,

acos(x)dx=xacos(x)1x2\int{\operatorname{acos}{\left(x \right)} d x} = x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}

Add the constant of integration:

acos(x)dx=xacos(x)1x2+C\int{\operatorname{acos}{\left(x \right)} d x} = x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}+C

Answer

acos(x)dx=(xacos(x)1x2)+C\int \operatorname{acos}{\left(x \right)}\, dx = \left(x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}\right) + CA