Mister Exam

Integral of xln(1+x) dx

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The solution

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01xlog(x+1)dx\int\limits_{0}^{1} x \log{\left(x + 1 \right)}\, dx
Integral(x*log(1 + x), (x, 0, 1))
Detail solution
  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(x+1)u{\left(x \right)} = \log{\left(x + 1 \right)} and let dv(x)=x\operatorname{dv}{\left(x \right)} = x.

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

    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:

      xdx=x22\int x\, dx = \frac{x^{2}}{2}

    Now evaluate the sub-integral.

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

    x22(x+1)dx=x2x+1dx2\int \frac{x^{2}}{2 \left(x + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x + 1}\, dx}{2}

    1. Rewrite the integrand:

      x2x+1=x1+1x+1\frac{x^{2}}{x + 1} = x - 1 + \frac{1}{x + 1}

    2. Integrate term-by-term:

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

        xdx=x22\int x\, dx = \frac{x^{2}}{2}

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

        (1)dx=x\int \left(-1\right)\, dx = - x

      1. Let u=x+1u = x + 1.

        Then let du=dxdu = dx and substitute dudu:

        1udu\int \frac{1}{u}\, du

        1. The integral of 1u\frac{1}{u} is log(u)\log{\left(u \right)}.

        Now substitute uu back in:

        log(x+1)\log{\left(x + 1 \right)}

      The result is: x22x+log(x+1)\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)}

    So, the result is: x24x2+log(x+1)2\frac{x^{2}}{4} - \frac{x}{2} + \frac{\log{\left(x + 1 \right)}}{2}

  3. Add the constant of integration:

    x2log(x+1)2x24+x2log(x+1)2+constant\frac{x^{2} \log{\left(x + 1 \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2} - \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}


The answer is:

x2log(x+1)2x24+x2log(x+1)2+constant\frac{x^{2} \log{\left(x + 1 \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2} - \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                        2    2           
 |                       x   log(1 + x)   x    x *log(1 + x)
 | x*log(1 + x) dx = C + - - ---------- - -- + -------------
 |                       2       2        4          2      
/                                                           
xlog(x+1)dx=C+x2log(x+1)2x24+x2log(x+1)2\int x \log{\left(x + 1 \right)}\, dx = C + \frac{x^{2} \log{\left(x + 1 \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2} - \frac{\log{\left(x + 1 \right)}}{2}
The graph
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The answer [src]
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14\frac{1}{4}
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14\frac{1}{4}
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Numerical answer [src]
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The graph
Integral of xln(1+x) dx

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