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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 n1n \neq -1:

        x2dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

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

        y2dx=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]
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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)
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.