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Solving integrals/ arcsin2x

Integral of arcsin2x dx

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The solution

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∫014asin⁡(2x) dx\int\limits_{0}^{\frac{1}{4}} \operatorname{asin}{\left(2 x \right)}\, dx 
Integral(asin(2*x), (x, 0, 1/4))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 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:

      xasin⁡(2x)+1−4x22x \operatorname{asin}{\left(2 x \right)} + \frac{\sqrt{1 - 4 x^{2}}}{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⁡(2x)u{\left(x \right)} = \operatorname{asin}{\left(2 x \right)} and let dv⁡(x)=1\operatorname{dv}{\left(x \right)} = 1.

      Then du⁡(x)=21−4x2\operatorname{du}{\left(x \right)} = \frac{2}{\sqrt{1 - 4 x^{2}}}.

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

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

        ∫1 dx=x\int 1\, dx = x

      Now evaluate the sub-integral.

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

      ∫2x1−4x2 dx=2∫x1−4x2 dx\int \frac{2 x}{\sqrt{1 - 4 x^{2}}}\, dx = 2 \int \frac{x}{\sqrt{1 - 4 x^{2}}}\, dx

      1. Let u=1−4x2u = 1 - 4 x^{2}.

        Then let du=−8xdxdu = - 8 x dx and substitute −du8- \frac{du}{8}:

        ∫(−18u) du\int \left(- \frac{1}{8 \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 du8\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{8}

          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: −u4- \frac{\sqrt{u}}{4}

        Now substitute uu back in:

        −1−4x24- \frac{\sqrt{1 - 4 x^{2}}}{4}

      So, the result is: −1−4x22- \frac{\sqrt{1 - 4 x^{2}}}{2}

  2. Add the constant of integration:

    xasin⁡(2x)+1−4x22+constantx \operatorname{asin}{\left(2 x \right)} + \frac{\sqrt{1 - 4 x^{2}}}{2}+ \mathrm{constant}

The answer is:

xasin⁡(2x)+1−4x22+constantx \operatorname{asin}{\left(2 x \right)} + \frac{\sqrt{1 - 4 x^{2}}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
                         __________              
  /                     /        2               
 |                    \/  1 - 4*x                
 | asin(2*x) dx = C + ------------- + x*asin(2*x)
 |                          2                    
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∫asin⁡(2x) dx=C+xasin⁡(2x)+1−4x22\int \operatorname{asin}{\left(2 x \right)}\, dx = C + x \operatorname{asin}{\left(2 x \right)} + \frac{\sqrt{1 - 4 x^{2}}}{2}
Simplify
The graph
0.0000.2500.0250.0500.0750.1000.1250.1500.1750.2000.2250.01.0
Plot the graph f(x)
The answer [src] 
        ___     
  1   \/ 3    pi
- - + ----- + --
  2     4     24
−12+π24+34- \frac{1}{2} + \frac{\pi}{24} + \frac{\sqrt{3}}{4} 
=
= 
        ___     
  1   \/ 3    pi
- - + ----- + --
  2     4     24
−12+π24+34- \frac{1}{2} + \frac{\pi}{24} + \frac{\sqrt{3}}{4} 
-1/2 + sqrt(3)/4 + pi/24
Numerical answer [src] 
0.063912395791794
0.063912395791794

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

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