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(x^4+3)*(4x^3dx)
Solving integrals/ (x^4+3)*(4x^3dx)

Integral of (x^4+3)*(4x^3dx) dx

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∫01(x4+3)4x3⋅1 dx\int\limits_{0}^{1} \left(x^{4} + 3\right) 4 x^{3} \cdot 1\, dx 
Integral((x^4 + 3)*4*x^3*1, (x, 0, 1))
Detail solution

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

    ∫(x4+3)4x3⋅1 dx=4∫x3(x4+3) dx\int \left(x^{4} + 3\right) 4 x^{3} \cdot 1\, dx = 4 \int x^{3} \left(x^{4} + 3\right)\, dx

    1. There are multiple ways to do this integral.

      Method #1

      1. Let u=x4u = x^{4}.

        Then let du=4x3dxdu = 4 x^{3} dx and substitute dudu:

        ∫(u4+34) du\int \left(\frac{u}{4} + \frac{3}{4}\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:

            ∫u4 du=∫u du4\int \frac{u}{4}\, du = \frac{\int u\, du}{4}

            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: u28\frac{u^{2}}{8}

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

            ∫34 du=3u4\int \frac{3}{4}\, du = \frac{3 u}{4}

          The result is: u28+3u4\frac{u^{2}}{8} + \frac{3 u}{4}

        Now substitute uu back in:

        x88+3x44\frac{x^{8}}{8} + \frac{3 x^{4}}{4}

      Method #2

      1. Rewrite the integrand:

        x3(x4+3)=x7+3x3x^{3} \left(x^{4} + 3\right) = x^{7} + 3 x^{3}

      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:

          ∫x7 dx=x88\int x^{7}\, dx = \frac{x^{8}}{8}

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

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

          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}

          So, the result is: 3x44\frac{3 x^{4}}{4}

        The result is: x88+3x44\frac{x^{8}}{8} + \frac{3 x^{4}}{4}

    So, the result is: x82+3x4\frac{x^{8}}{2} + 3 x^{4}

  2. Now simplify:

    x4(x4+6)2\frac{x^{4} \left(x^{4} + 6\right)}{2}

  3. Add the constant of integration:

    x4(x4+6)2+constant\frac{x^{4} \left(x^{4} + 6\right)}{2}+ \mathrm{constant}

The answer is:

x4(x4+6)2+constant\frac{x^{4} \left(x^{4} + 6\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | \x  + 3/*4*x *1 dx = C + -- + 3*x 
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(x4+3)22{{\left(x^4+3\right)^2}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90020
Plot the graph f(x)
The answer [src] 
7/2
72{{7}\over{2}} 
=
= 
7/2
72\frac{7}{2}
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Numerical answer [src] 
3.5
3.5
The graph
Integral of (x^4+3)*(4x^3dx) dx

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

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