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Solving integrals/ (2xy^2-2x^3)

Integral of (2xy^2-2x^3) dx

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∫01(−2x3+2xy2) dx\int\limits_{0}^{1} \left(- 2 x^{3} + 2 x y^{2}\right)\, dx 
Integral(2*x*y^2 - 2*x^3, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

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

      ∫(−2x3) dx=−∫2x3 dx\int \left(- 2 x^{3}\right)\, dx = - \int 2 x^{3}\, dx

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

        ∫2x3 dx=2∫x3 dx\int 2 x^{3}\, dx = 2 \int x^{3}\, dx

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

          ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

        So, the result is: x42\frac{x^{4}}{2}

      So, the result is: −x42- \frac{x^{4}}{2}

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

      ∫2xy2 dx=2y2∫x dx\int 2 x y^{2}\, dx = 2 y^{2} \int x\, dx

      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}

      So, the result is: x2y2x^{2} y^{2}

    The result is: −x42+x2y2- \frac{x^{4}}{2} + x^{2} y^{2}

  2. Now simplify:

    x2(−x22+y2)x^{2} \left(- \frac{x^{2}}{2} + y^{2}\right)

  3. Add the constant of integration:

    x2(−x22+y2)+constantx^{2} \left(- \frac{x^{2}}{2} + y^{2}\right)+ \mathrm{constant}

The answer is:

x2(−x22+y2)+constantx^{2} \left(- \frac{x^{2}}{2} + y^{2}\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | \2*x*y  - 2*x / dx = C - -- + x *y 
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x2 y2−x42x^2\,y^2-{{x^4}\over{2}}
Simplify
The answer [src] 
  1    2
- - + y 
  2
2 y2−12{{2\,y^2-1}\over{2}} 
=
= 
  1    2
- - + y 
  2
y2−12y^{2} - \frac{1}{2}
Simplify

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

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