Mister Exam

Other calculators

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

Integral of log(x)*dx/x^3 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1          
  /          
 |           
 |  log(x)   
 |  ------ dx
 |     3     
 |    x      
 |           
/            
0
∫01log⁡(x)x3 dx\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{x^{3}}\, dx 
Integral(log(x)/x^3, (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:

      ∫ue−2u du\int u 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)=uu{\left(u \right)} = u and let dv⁡(u)=e−2u\operatorname{dv}{\left(u \right)} = e^{- 2 u}.

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

        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 \left(- \frac{e^{u}}{2}\right)\, 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:

          −e−2u2- \frac{e^{- 2 u}}{2}

        Now evaluate the sub-integral.

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

        ∫(−e−2u2) du=−∫e−2u du2\int \left(- \frac{e^{- 2 u}}{2}\right)\, du = - \frac{\int e^{- 2 u}\, du}{2}

        1. Let u=−2uu = - 2 u.

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

          ∫(−eu2) du\int \left(- \frac{e^{u}}{2}\right)\, 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:

          −e−2u2- \frac{e^{- 2 u}}{2}

        So, the result is: e−2u4\frac{e^{- 2 u}}{4}

      Now substitute uu back in:

      −log⁡(x)2x2−14x2- \frac{\log{\left(x \right)}}{2 x^{2}} - \frac{1}{4 x^{2}}

    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)=1x3\operatorname{dv}{\left(x \right)} = \frac{1}{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:

        ∫1x3 dx=−12x2\int \frac{1}{x^{3}}\, dx = - \frac{1}{2 x^{2}}

      Now evaluate the sub-integral.

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

      ∫(−12x3) dx=−∫1x3 dx2\int \left(- \frac{1}{2 x^{3}}\right)\, dx = - \frac{\int \frac{1}{x^{3}}\, dx}{2}

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫1x3 dx=−12x2\int \frac{1}{x^{3}}\, dx = - \frac{1}{2 x^{2}}

      So, the result is: 14x2\frac{1}{4 x^{2}}

  2. Now simplify:

    −2log⁡(x)+14x2- \frac{2 \log{\left(x \right)} + 1}{4 x^{2}}

  3. Add the constant of integration:

    −2log⁡(x)+14x2+constant- \frac{2 \log{\left(x \right)} + 1}{4 x^{2}}+ \mathrm{constant}

The answer is:

−2log⁡(x)+14x2+constant- \frac{2 \log{\left(x \right)} + 1}{4 x^{2}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                             
 |                              
 | log(x)           1     log(x)
 | ------ dx = C - ---- - ------
 |    3               2       2 
 |   x             4*x     2*x  
 |                              
/
∫log⁡(x)x3 dx=C−log⁡(x)2x2−14x2\int \frac{\log{\left(x \right)}}{x^{3}}\, dx = C - \frac{\log{\left(x \right)}}{2 x^{2}} - \frac{1}{4 x^{2}}
Simplify
The answer [src] 
-oo
−∞-\infty 
=
= 
-oo
−∞-\infty 
-oo
Numerical answer [src] 
-3.98309783194748e+39
-3.98309783194748e+39

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

    © Mister Exam – Calculator