Mister Exam

Other calculators

x*arccos(x)
Solving integrals/ x*arccos(x)

Integral of x*arccos(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |  x*acos(x) dx
 |              
/               
0
∫01xacos⁡(x) dx\int\limits_{0}^{1} x \operatorname{acos}{\left(x \right)}\, dx
Simplify
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

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

    Then du⁡(x)=−11−x2\operatorname{du}{\left(x \right)} = - \frac{1}{\sqrt{1 - x^{2}}}.

    To find v(x)v{\left(x \right)}:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫(−x221−x2) dx=−∫x21−x2 dx2\int \left(- \frac{x^{2}}{2 \sqrt{1 - x^{2}}}\right)\, dx = - \frac{\int \frac{x^{2}}{\sqrt{1 - x^{2}}}\, dx}{2}

      SqrtQuadraticDenomRule(a=1, b=0, c=-1, coeffs=[1, 0, 0], context=x**2/sqrt(1 - x**2), symbol=x)

    So, the result is: x1−x24−asin⁡(x)4\frac{x \sqrt{1 - x^{2}}}{4} - \frac{\operatorname{asin}{\left(x \right)}}{4}

  3. Add the constant of integration:

    x2acos⁡(x)2−x1−x24+asin⁡(x)4+constant\frac{x^{2} \operatorname{acos}{\left(x \right)}}{2} - \frac{x \sqrt{1 - x^{2}}}{4} + \frac{\operatorname{asin}{\left(x \right)}}{4}+ \mathrm{constant}

The answer is:

x2acos⁡(x)2−x1−x24+asin⁡(x)4+constant\frac{x^{2} \operatorname{acos}{\left(x \right)}}{2} - \frac{x \sqrt{1 - x^{2}}}{4} + \frac{\operatorname{asin}{\left(x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
                                                  ________
  /                              2               /      2 
 |                    asin(x)   x *acos(x)   x*\/  1 - x  
 | x*acos(x) dx = C + ------- + ---------- - -------------
 |                       4          2              4      
/
arcsin⁡x2−x 1−x222+x2 arccos⁡x2{{{{\arcsin x}\over{2}}-{{x\,\sqrt{1-x^2}}\over{2}}}\over{2}}+{{x^2 \,\arccos x}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-1
Plot the graph f(x)
The answer [src] 
pi
--
8
π8{{\pi}\over{8}} 
=
= 
pi
--
8
π8\frac{\pi}{8}
Simplify
Numerical answer [src] 
0.392699081698724
0.392699081698724
The graph
Integral of x*arccos(x) dx

    Use the examples entering the upper and lower limits of integration.

    © Mister Exam – Calculator