Mister Exam

Other calculators

Solving integrals/ x^3+xy^2

Integral of x^3+xy^2 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1               
  /               
 |                
 |  / 3      2\   
 |  \x  + x*y / dx
 |                
/                 
0
∫01(x3+xy2) dx\int\limits_{0}^{1} \left(x^{3} + x y^{2}\right)\, dx 
Integral(x^3 + x*y^2, (x, 0, 1))
Detail solution

  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:

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

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

      ∫xy2 dx=y2∫x dx\int x y^{2}\, dx = 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: x2y22\frac{x^{2} y^{2}}{2}

    The result is: x44+x2y22\frac{x^{4}}{4} + \frac{x^{2} y^{2}}{2}

  2. Now simplify:

    x2(x2+2y2)4\frac{x^{2} \left(x^{2} + 2 y^{2}\right)}{4}

  3. Add the constant of integration:

    x2(x2+2y2)4+constant\frac{x^{2} \left(x^{2} + 2 y^{2}\right)}{4}+ \mathrm{constant}

The answer is:

x2(x2+2y2)4+constant\frac{x^{2} \left(x^{2} + 2 y^{2}\right)}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
 |                       4    2  2
 | / 3      2\          x    x *y 
 | \x  + x*y / dx = C + -- + -----
 |                      4      2  
/
x2 y22+x44{{x^2\,y^2}\over{2}}+{{x^4}\over{4}}
Simplify
The answer [src] 
     2
1   y 
- + --
4   2
2 y2+14{{2\,y^2+1}\over{4}} 
=
= 
     2
1   y 
- + --
4   2
y22+14\frac{y^{2}}{2} + \frac{1}{4}
Simplify

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

    © Mister Exam – Calculator