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Solving integrals/ x*asin(x^2)

Integral of x*asin(x^2) dx

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∫01xasin⁡(x2) dx\int\limits_{0}^{1} x \operatorname{asin}{\left(x^{2} \right)}\, dx 
Integral(x*asin(x^2), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x2u = x^{2}.

      Then let du=2xdxdu = 2 x dx and substitute du2\frac{du}{2}:

      ∫asin⁡(u)2 du\int \frac{\operatorname{asin}{\left(u \right)}}{2}\, du

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

        ∫asin⁡(u) du=∫asin⁡(u) du2\int \operatorname{asin}{\left(u \right)}\, du = \frac{\int \operatorname{asin}{\left(u \right)}\, du}{2}

        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(u)=asin⁡(u)u{\left(u \right)} = \operatorname{asin}{\left(u \right)} and let dv⁡(u)=1\operatorname{dv}{\left(u \right)} = 1.

          Then du⁡(u)=11−u2\operatorname{du}{\left(u \right)} = \frac{1}{\sqrt{1 - u^{2}}}.

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

          1. The integral of a constant is the constant times the variable of integration:

            ∫1 du=u\int 1\, du = u

          Now evaluate the sub-integral.

        2. Let u=1−u2u = 1 - u^{2}.

          Then let du=−2ududu = - 2 u du and substitute −du2- \frac{du}{2}:

          ∫(−12u) du\int \left(- \frac{1}{2 \sqrt{u}}\right)\, du

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

            ∫1u du=−∫1u du2\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{2}

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫1u du=2u\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}

            So, the result is: −u- \sqrt{u}

          Now substitute uu back in:

          −1−u2- \sqrt{1 - u^{2}}

        So, the result is: uasin⁡(u)2+1−u22\frac{u \operatorname{asin}{\left(u \right)}}{2} + \frac{\sqrt{1 - u^{2}}}{2}

      Now substitute uu back in:

      x2asin⁡(x2)2+1−x42\frac{x^{2} \operatorname{asin}{\left(x^{2} \right)}}{2} + \frac{\sqrt{1 - x^{4}}}{2}

    Method #2

    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)=asin⁡(x2)u{\left(x \right)} = \operatorname{asin}{\left(x^{2} \right)} and let dv⁡(x)=x\operatorname{dv}{\left(x \right)} = x.

      Then du⁡(x)=2x1−x4\operatorname{du}{\left(x \right)} = \frac{2 x}{\sqrt{1 - x^{4}}}.

      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. Let u=1−x4u = 1 - x^{4}.

      Then let du=−4x3dxdu = - 4 x^{3} dx and substitute −du4- \frac{du}{4}:

      ∫(−14u) du\int \left(- \frac{1}{4 \sqrt{u}}\right)\, du

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

        ∫1u du=−∫1u du4\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{4}

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫1u du=2u\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}

        So, the result is: −u2- \frac{\sqrt{u}}{2}

      Now substitute uu back in:

      −1−x42- \frac{\sqrt{1 - x^{4}}}{2}

  2. Add the constant of integration:

    x2asin⁡(x2)2+1−x42+constant\frac{x^{2} \operatorname{asin}{\left(x^{2} \right)}}{2} + \frac{\sqrt{1 - x^{4}}}{2}+ \mathrm{constant}

The answer is:

x2asin⁡(x2)2+1−x42+constant\frac{x^{2} \operatorname{asin}{\left(x^{2} \right)}}{2} + \frac{\sqrt{1 - x^{4}}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                       ________              
 |                       /      4     2     / 2\
 |       / 2\          \/  1 - x     x *asin\x /
 | x*asin\x / dx = C + ----------- + -----------
 |                          2             2     
/
∫xasin⁡(x2) dx=C+x2asin⁡(x2)2+1−x42\int x \operatorname{asin}{\left(x^{2} \right)}\, dx = C + \frac{x^{2} \operatorname{asin}{\left(x^{2} \right)}}{2} + \frac{\sqrt{1 - x^{4}}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
  1   pi
- - + --
  2   4
−12+π4- \frac{1}{2} + \frac{\pi}{4} 
=
= 
  1   pi
- - + --
  2   4
−12+π4- \frac{1}{2} + \frac{\pi}{4} 
-1/2 + pi/4
Numerical answer [src] 
0.285398163397448
0.285398163397448

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

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