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Solving integrals/ y(x^2+y^2)

Integral of y(x^2+y^2) dx

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The solution

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∫01y(x2+y2) dx\int\limits_{0}^{1} y \left(x^{2} + y^{2}\right)\, dx 
Integral(y*(x^2 + y^2), (x, 0, 1))
Detail solution

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

    ∫y(x2+y2) dx=y∫(x2+y2) dx\int y \left(x^{2} + y^{2}\right)\, dx = y \int \left(x^{2} + y^{2}\right)\, dx

    1. Integrate term-by-term:

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

        ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

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

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

      The result is: x33+xy2\frac{x^{3}}{3} + x y^{2}

    So, the result is: y(x33+xy2)y \left(\frac{x^{3}}{3} + x y^{2}\right)

  2. Now simplify:

    xy(x23+y2)x y \left(\frac{x^{2}}{3} + y^{2}\right)

  3. Add the constant of integration:

    xy(x23+y2)+constantx y \left(\frac{x^{2}}{3} + y^{2}\right)+ \mathrm{constant}

The answer is:

xy(x23+y2)+constantx y \left(\frac{x^{2}}{3} + y^{2}\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                        / 3       \
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 | y*\x  + y / dx = C + y*|-- + x*y |
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∫y(x2+y2) dx=C+y(x33+xy2)\int y \left(x^{2} + y^{2}\right)\, dx = C + y \left(\frac{x^{3}}{3} + x y^{2}\right)
Simplify
The answer [src] 
 3   y
y  + -
     3
y3+y3y^{3} + \frac{y}{3} 
=
= 
 3   y
y  + -
     3
y3+y3y^{3} + \frac{y}{3} 
y^3 + y/3

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

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