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Solving integrals/ 2x+y^2

Integral of 2x+y^2 dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
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∫y3y2(2x+y2) dx\int\limits_{y^{3}}^{y^{2}} \left(2 x + y^{2}\right)\, dx 
Integral(2*x + y^2, (x, y^3, y^2))
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:

      ∫2x dx=2∫x dx\int 2 x\, dx = 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: x2x^{2}

    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: x2+xy2x^{2} + x y^{2}

  2. Now simplify:

    x(x+y2)x \left(x + y^{2}\right)

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                             
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 | /       2\           2      2
 | \2*x + y / dx = C + x  + x*y 
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∫(2x+y2) dx=C+x2+xy2\int \left(2 x + y^{2}\right)\, dx = C + x^{2} + x y^{2}
Simplify
The answer [src] 
   5    6      4
- y  - y  + 2*y
−y6−y5+2y4- y^{6} - y^{5} + 2 y^{4} 
=
= 
   5    6      4
- y  - y  + 2*y
−y6−y5+2y4- y^{6} - y^{5} + 2 y^{4} 
-y^5 - y^6 + 2*y^4

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

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