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log(2x^5)/x^2

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log(2x^5)/x^2

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Integral of log(2x^5)/x^2 dx

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  3             
  /             
 |              
 |     /   5\   
 |  log\2*x /   
 |  --------- dx
 |       2      
 |      x       
 |              
/               
2               
23log(2x5)x2dx\int\limits_{2}^{3} \frac{\log{\left(2 x^{5} \right)}}{x^{2}}\, dx
Integral(log(2*x^5)/(x^2), (x, 2, 3))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      log(2x5)x2=log(x5)+log(2)x2\frac{\log{\left(2 x^{5} \right)}}{x^{2}} = \frac{\log{\left(x^{5} \right)} + \log{\left(2 \right)}}{x^{2}}

    2. Rewrite the integrand:

      log(x5)+log(2)x2=log(x5)x2+log(2)x2\frac{\log{\left(x^{5} \right)} + \log{\left(2 \right)}}{x^{2}} = \frac{\log{\left(x^{5} \right)}}{x^{2}} + \frac{\log{\left(2 \right)}}{x^{2}}

    3. Integrate term-by-term:

      1. Use integration by parts:

        udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

        Let u(x)=log(x5)u{\left(x \right)} = \log{\left(x^{5} \right)} and let dv(x)=1x2\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}.

        Then du(x)=5x\operatorname{du}{\left(x \right)} = \frac{5}{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 n1n \neq -1:

          1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

        Now evaluate the sub-integral.

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

        (5x2)dx=51x2dx\int \left(- \frac{5}{x^{2}}\right)\, dx = - 5 \int \frac{1}{x^{2}}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

        So, the result is: 5x\frac{5}{x}

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

        log(2)x2dx=log(2)1x2dx\int \frac{\log{\left(2 \right)}}{x^{2}}\, dx = \log{\left(2 \right)} \int \frac{1}{x^{2}}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

        So, the result is: log(2)x- \frac{\log{\left(2 \right)}}{x}

      The result is: log(x5)x5xlog(2)x- \frac{\log{\left(x^{5} \right)}}{x} - \frac{5}{x} - \frac{\log{\left(2 \right)}}{x}

    Method #2

    1. Use integration by parts:

      udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log(2x5)u{\left(x \right)} = \log{\left(2 x^{5} \right)} and let dv(x)=1x2\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}.

      Then du(x)=5x\operatorname{du}{\left(x \right)} = \frac{5}{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 n1n \neq -1:

        1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

      Now evaluate the sub-integral.

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

      (5x2)dx=51x2dx\int \left(- \frac{5}{x^{2}}\right)\, dx = - 5 \int \frac{1}{x^{2}}\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

        1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

      So, the result is: 5x\frac{5}{x}

    Method #3

    1. Rewrite the integrand:

      log(2x5)x2=log(x5)x2+log(2)x2\frac{\log{\left(2 x^{5} \right)}}{x^{2}} = \frac{\log{\left(x^{5} \right)}}{x^{2}} + \frac{\log{\left(2 \right)}}{x^{2}}

    2. Integrate term-by-term:

      1. Use integration by parts:

        udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

        Let u(x)=log(x5)u{\left(x \right)} = \log{\left(x^{5} \right)} and let dv(x)=1x2\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}.

        Then du(x)=5x\operatorname{du}{\left(x \right)} = \frac{5}{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 n1n \neq -1:

          1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

        Now evaluate the sub-integral.

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

        (5x2)dx=51x2dx\int \left(- \frac{5}{x^{2}}\right)\, dx = - 5 \int \frac{1}{x^{2}}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

        So, the result is: 5x\frac{5}{x}

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

        log(2)x2dx=log(2)1x2dx\int \frac{\log{\left(2 \right)}}{x^{2}}\, dx = \log{\left(2 \right)} \int \frac{1}{x^{2}}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

        So, the result is: log(2)x- \frac{\log{\left(2 \right)}}{x}

      The result is: log(x5)x5xlog(2)x- \frac{\log{\left(x^{5} \right)}}{x} - \frac{5}{x} - \frac{\log{\left(2 \right)}}{x}

  2. Now simplify:

    log(x5)5log(2)x\frac{- \log{\left(x^{5} \right)} - 5 - \log{\left(2 \right)}}{x}

  3. Add the constant of integration:

    log(x5)5log(2)x+constant\frac{- \log{\left(x^{5} \right)} - 5 - \log{\left(2 \right)}}{x}+ \mathrm{constant}


The answer is:

log(x5)5log(2)x+constant\frac{- \log{\left(x^{5} \right)} - 5 - \log{\left(2 \right)}}{x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                       
 |                                        
 |    /   5\                          / 5\
 | log\2*x /          5   log(2)   log\x /
 | --------- dx = C - - - ------ - -------
 |      2             x     x         x   
 |     x                                  
 |                                        
/                                         
log(2x5)x5x-{{\log \left(2\,x^5\right)}\over{x}}-{{5}\over{x}}
The graph
2.003.002.102.202.302.402.502.602.702.802.905-5
The answer [src]
5   log(64)   log(486)
- + ------- - --------
6      2         3    
215(5log4863215+5log64265253215+25265)5{{2^{{{1}\over{5}}}\,\left(-{{5\,\log 486}\over{3\,2^{{{1}\over{5}} }}}+{{5\,\log 64}\over{2^{{{6}\over{5}}}}}-{{25}\over{3\,2^{{{1 }\over{5}}}}}+{{25}\over{2^{{{6}\over{5}}}}}\right)}\over{5}}
=
=
5   log(64)   log(486)
- + ------- - --------
6      2         3    
log(486)3+56+log(64)2- \frac{\log{\left(486 \right)}}{3} + \frac{5}{6} + \frac{\log{\left(64 \right)}}{2}
Numerical answer [src]
0.850705333713005
0.850705333713005
The graph
Integral of log(2x^5)/x^2 dx

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