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2x*ln(x+1)

Integral of 2x*ln(x+1) dx

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The solution

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012xlog(x+1)dx\int\limits_{0}^{1} 2 x \log{\left(x + 1 \right)}\, dx
Integral((2*x)*log(x + 1), (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)=2x\operatorname{dv}{\left(x \right)} = 2 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 a constant times a function is the constant times the integral of the function:

      2xdx=2xdx\int 2 x\, dx = 2 \int x\, dx

      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}

      So, the result is: x2x^{2}

    Now evaluate the sub-integral.

  2. Rewrite the integrand:

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

  3. 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)}

  4. Now simplify:

    x2log(x+1)x22+xlog(x+1)x^{2} \log{\left(x + 1 \right)} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}

  5. Add the constant of integration:

    x2log(x+1)x22+xlog(x+1)+constantx^{2} \log{\left(x + 1 \right)} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}+ \mathrm{constant}


The answer is:

x2log(x+1)x22+xlog(x+1)+constantx^{2} \log{\left(x + 1 \right)} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
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 | 2*x*log(x + 1) dx = C + x - log(1 + x) - -- + x *log(x + 1)
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2xlog(x+1)dx=C+x2log(x+1)x22+xlog(x+1)\int 2 x \log{\left(x + 1 \right)}\, dx = C + x^{2} \log{\left(x + 1 \right)} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}
The graph
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The answer [src]
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12\frac{1}{2}
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12\frac{1}{2}
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Numerical answer [src]
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The graph
Integral of 2x*ln(x+1) dx

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