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x+lnx
Solving integrals/ x+lnx

Integral of x+lnx dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

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∫01(x+log⁡(x)) dx\int\limits_{0}^{1} \left(x + \log{\left(x \right)}\right)\, dx 
Integral(x + log(x), (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

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

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{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)=1\operatorname{dv}{\left(x \right)} = 1.

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

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

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

        ∫1 dx=x\int 1\, dx = x

      Now evaluate the sub-integral.

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

      ∫1 dx=x\int 1\, dx = x

    The result is: x22+xlog⁡(x)−x\frac{x^{2}}{2} + x \log{\left(x \right)} - x

  2. Now simplify:

    x(x+2log⁡(x)−2)2\frac{x \left(x + 2 \log{\left(x \right)} - 2\right)}{2}

  3. Add the constant of integration:

    x(x+2log⁡(x)−2)2+constant\frac{x \left(x + 2 \log{\left(x \right)} - 2\right)}{2}+ \mathrm{constant}

The answer is:

x(x+2log⁡(x)−2)2+constant\frac{x \left(x + 2 \log{\left(x \right)} - 2\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫(x+log⁡(x)) dx=C+x22+xlog⁡(x)−x\int \left(x + \log{\left(x \right)}\right)\, dx = C + \frac{x^{2}}{2} + x \log{\left(x \right)} - x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1010
Plot the graph f(x)
The answer [src] 
-1/2
−12- \frac{1}{2} 
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= 
-1/2
−12- \frac{1}{2} 
-1/2
Numerical answer [src] 
-0.5
-0.5
The graph
Integral of x+lnx dx

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

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