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x^3*log(x)
Solving integrals/ x^3*log(x)

Integral of x^3*log(x) dx

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The solution

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∫01x3log⁡(x) dx\int\limits_{0}^{1} x^{3} \log{\left(x \right)}\, dx 
Integral(x^3*log(x), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=log⁡(x)u = \log{\left(x \right)}.

      Then let du=dxxdu = \frac{dx}{x} and substitute dudu:

      ∫ue4u du\int u e^{4 u}\, du

      1. Use integration by parts:

        ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

        Let u(u)=uu{\left(u \right)} = u and let dv⁡(u)=e4u\operatorname{dv}{\left(u \right)} = e^{4 u}.

        Then du⁡(u)=1\operatorname{du}{\left(u \right)} = 1.

        To find v(u)v{\left(u \right)}:

        1. Let u=4uu = 4 u.

          Then let du=4dudu = 4 du and substitute du4\frac{du}{4}:

          ∫eu4 du\int \frac{e^{u}}{4}\, du

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

            False\text{False}

            1. The integral of the exponential function is itself.

              ∫eu du=eu\int e^{u}\, du = e^{u}

            So, the result is: eu4\frac{e^{u}}{4}

          Now substitute uu back in:

          e4u4\frac{e^{4 u}}{4}

        Now evaluate the sub-integral.

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

        ∫e4u4 du=∫e4u du4\int \frac{e^{4 u}}{4}\, du = \frac{\int e^{4 u}\, du}{4}

        1. Let u=4uu = 4 u.

          Then let du=4dudu = 4 du and substitute du4\frac{du}{4}:

          ∫eu4 du\int \frac{e^{u}}{4}\, du

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

            False\text{False}

            1. The integral of the exponential function is itself.

              ∫eu du=eu\int e^{u}\, du = e^{u}

            So, the result is: eu4\frac{e^{u}}{4}

          Now substitute uu back in:

          e4u4\frac{e^{4 u}}{4}

        So, the result is: e4u16\frac{e^{4 u}}{16}

      Now substitute uu back in:

      x4log⁡(x)4−x416\frac{x^{4} \log{\left(x \right)}}{4} - \frac{x^{4}}{16}

    Method #2

    1. Use integration by parts:

      ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log⁡(x)u{\left(x \right)} = \log{\left(x \right)} and let dv⁡(x)=x3\operatorname{dv}{\left(x \right)} = x^{3}.

      Then du⁡(x)=1x\operatorname{du}{\left(x \right)} = \frac{1}{x}.

      To find v(x)v{\left(x \right)}:

      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}

      Now evaluate the sub-integral.

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

      ∫x34 dx=∫x3 dx4\int \frac{x^{3}}{4}\, dx = \frac{\int x^{3}\, dx}{4}

      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: x416\frac{x^{4}}{16}

  2. Now simplify:

    x4(4log⁡(x)−1)16\frac{x^{4} \left(4 \log{\left(x \right)} - 1\right)}{16}

  3. Add the constant of integration:

    x4(4log⁡(x)−1)16+constant\frac{x^{4} \left(4 \log{\left(x \right)} - 1\right)}{16}+ \mathrm{constant}

The answer is:

x4(4log⁡(x)−1)16+constant\frac{x^{4} \left(4 \log{\left(x \right)} - 1\right)}{16}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
 |                     4    4       
 |  3                 x    x *log(x)
 | x *log(x) dx = C - -- + ---------
 |                    16       4    
/
∫x3log⁡(x) dx=C+x4log⁡(x)4−x416\int x^{3} \log{\left(x \right)}\, dx = C + \frac{x^{4} \log{\left(x \right)}}{4} - \frac{x^{4}}{16}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-0.20.1
Plot the graph f(x)
The answer [src] 
-1/16
−116- \frac{1}{16} 
=
= 
-1/16
−116- \frac{1}{16} 
-1/16
Numerical answer [src] 
-0.0625
-0.0625
The graph
Integral of x^3*log(x) dx

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

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