Mister Exam

Other calculators

Solving integrals/ x*log^3*x

Integral of x*log^3*x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |       3      
 |  x*log (x) dx
 |              
/               
0
∫01xlog⁡(x)3 dx\int\limits_{0}^{1} x \log{\left(x \right)}^{3}\, dx 
Integral(x*log(x)^3, (x, 0, 1))
Detail solution

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

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

    ∫u3e2u du\int u^{3} e^{2 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)=u3u{\left(u \right)} = u^{3} and let dv⁡(u)=e2u\operatorname{dv}{\left(u \right)} = e^{2 u}.

      Then du⁡(u)=3u2\operatorname{du}{\left(u \right)} = 3 u^{2}.

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

      1. Let u=2uu = 2 u.

        Then let du=2dudu = 2 du and substitute du2\frac{du}{2}:

        ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

        Now substitute uu back in:

        e2u2\frac{e^{2 u}}{2}

      Now evaluate the sub-integral.

    2. 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)=3u22u{\left(u \right)} = \frac{3 u^{2}}{2} and let dv⁡(u)=e2u\operatorname{dv}{\left(u \right)} = e^{2 u}.

      Then du⁡(u)=3u\operatorname{du}{\left(u \right)} = 3 u.

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

      1. Let u=2uu = 2 u.

        Then let du=2dudu = 2 du and substitute du2\frac{du}{2}:

        ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

        Now substitute uu back in:

        e2u2\frac{e^{2 u}}{2}

      Now evaluate the sub-integral.

    3. 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)=3u2u{\left(u \right)} = \frac{3 u}{2} and let dv⁡(u)=e2u\operatorname{dv}{\left(u \right)} = e^{2 u}.

      Then du⁡(u)=32\operatorname{du}{\left(u \right)} = \frac{3}{2}.

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

      1. Let u=2uu = 2 u.

        Then let du=2dudu = 2 du and substitute du2\frac{du}{2}:

        ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

        Now substitute uu back in:

        e2u2\frac{e^{2 u}}{2}

      Now evaluate the sub-integral.

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

      ∫3e2u4 du=3∫e2u du4\int \frac{3 e^{2 u}}{4}\, du = \frac{3 \int e^{2 u}\, du}{4}

      1. Let u=2uu = 2 u.

        Then let du=2dudu = 2 du and substitute du2\frac{du}{2}:

        ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

        Now substitute uu back in:

        e2u2\frac{e^{2 u}}{2}

      So, the result is: 3e2u8\frac{3 e^{2 u}}{8}

    Now substitute uu back in:

    x2log⁡(x)32−3x2log⁡(x)24+3x2log⁡(x)4−3x28\frac{x^{2} \log{\left(x \right)}^{3}}{2} - \frac{3 x^{2} \log{\left(x \right)}^{2}}{4} + \frac{3 x^{2} \log{\left(x \right)}}{4} - \frac{3 x^{2}}{8}

  2. Now simplify:

    x2(4log⁡(x)3−6log⁡(x)2+6log⁡(x)−3)8\frac{x^{2} \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 3\right)}{8}

  3. Add the constant of integration:

    x2(4log⁡(x)3−6log⁡(x)2+6log⁡(x)−3)8+constant\frac{x^{2} \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 3\right)}{8}+ \mathrm{constant}

The answer is:

x2(4log⁡(x)3−6log⁡(x)2+6log⁡(x)−3)8+constant\frac{x^{2} \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 3\right)}{8}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                 
 |                       2    2    3         2    2         2       
 |      3             3*x    x *log (x)   3*x *log (x)   3*x *log(x)
 | x*log (x) dx = C - ---- + ---------- - ------------ + -----------
 |                     8         2             4              4     
/
∫xlog⁡(x)3 dx=C+x2log⁡(x)32−3x2log⁡(x)24+3x2log⁡(x)4−3x28\int x \log{\left(x \right)}^{3}\, dx = C + \frac{x^{2} \log{\left(x \right)}^{3}}{2} - \frac{3 x^{2} \log{\left(x \right)}^{2}}{4} + \frac{3 x^{2} \log{\left(x \right)}}{4} - \frac{3 x^{2}}{8}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-2
Plot the graph f(x)
The answer [src] 
-3/8
−38- \frac{3}{8} 
=
= 
-3/8
−38- \frac{3}{8} 
-3/8
Numerical answer [src] 
-0.375
-0.375

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

    © Mister Exam – Calculator