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Solving integrals/ x^2-y

Integral of x^2-y dy

Limits of integration:

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The solution

You have entered [src] 
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∫02x3(x2−y) dy\int\limits_{0}^{2 x^{3}} \left(x^{2} - y\right)\, dy 
Integral(x^2 - y, (y, 0, 2*x^3))
Detail solution

  1. Integrate term-by-term:

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

      ∫x2 dy=x2y\int x^{2}\, dy = x^{2} y

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

      ∫(−y) dy=−∫y dy\int \left(- y\right)\, dy = - \int y\, dy

      1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫y dy=y22\int y\, dy = \frac{y^{2}}{2}

      So, the result is: −y22- \frac{y^{2}}{2}

    The result is: x2y−y22x^{2} y - \frac{y^{2}}{2}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                           
 |                    2       
 | / 2    \          y       2
 | \x  - y/ dy = C - -- + y*x 
 |                   2        
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∫(x2−y) dy=C+x2y−y22\int \left(x^{2} - y\right)\, dy = C + x^{2} y - \frac{y^{2}}{2}
Simplify
The answer [src] 
     6      5
- 2*x  + 2*x
−2x6+2x5- 2 x^{6} + 2 x^{5} 
=
= 
     6      5
- 2*x  + 2*x
−2x6+2x5- 2 x^{6} + 2 x^{5} 
-2*x^6 + 2*x^5

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

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