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x(x^2+3)
Solving integrals/ x(x^2+3)

Integral of x(x^2+3) dx

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The solution

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∫01x(x2+3) dx\int\limits_{0}^{1} x \left(x^{2} + 3\right)\, dx 
Integral(x*(x^2 + 3), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x2u = x^{2}.

      Then let du=2xdxdu = 2 x dx and substitute dudu:

      ∫(u2+32) du\int \left(\frac{u}{2} + \frac{3}{2}\right)\, du

      1. Integrate term-by-term:

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

          ∫u2 du=∫u du2\int \frac{u}{2}\, du = \frac{\int u\, du}{2}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u du=u22\int u\, du = \frac{u^{2}}{2}

          So, the result is: u24\frac{u^{2}}{4}

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

          ∫32 du=3u2\int \frac{3}{2}\, du = \frac{3 u}{2}

        The result is: u24+3u2\frac{u^{2}}{4} + \frac{3 u}{2}

      Now substitute uu back in:

      x44+3x22\frac{x^{4}}{4} + \frac{3 x^{2}}{2}

    Method #2

    1. Rewrite the integrand:

      x(x2+3)=x3+3xx \left(x^{2} + 3\right) = x^{3} + 3 x

    2. 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:

        ∫3x dx=3∫x dx\int 3 x\, dx = 3 \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: 3x22\frac{3 x^{2}}{2}

      The result is: x44+3x22\frac{x^{4}}{4} + \frac{3 x^{2}}{2}

  2. Now simplify:

    x2(x2+6)4\frac{x^{2} \left(x^{2} + 6\right)}{4}

  3. Add the constant of integration:

    x2(x2+6)4+constant\frac{x^{2} \left(x^{2} + 6\right)}{4}+ \mathrm{constant}

The answer is:

x2(x2+6)4+constant\frac{x^{2} \left(x^{2} + 6\right)}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                             
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 | x*\x  + 3/ dx = C + -- + ----
 |                     4     2  
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∫x(x2+3) dx=C+x44+3x22\int x \left(x^{2} + 3\right)\, dx = C + \frac{x^{4}}{4} + \frac{3 x^{2}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9005
Plot the graph f(x)
The answer [src] 
7/4
74\frac{7}{4} 
=
= 
7/4
74\frac{7}{4} 
7/4
Numerical answer [src] 
1.75
1.75
The graph
Integral of x(x^2+3) dx

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

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