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x*ln(x+1)

Integral of x*ln(x+1) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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 |  x*log(x + 1) dx
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$$\int\limits_{0}^{1} x \log{\left(x + 1 \right)}\, dx$$
Integral(x*log(x + 1), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of is when :

    Now evaluate the sub-integral.

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

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of is when :

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

      1. Let .

        Then let and substitute :

        1. The integral of is .

        Now substitute back in:

      The result is:

    So, the result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                        2    2           
 |                       x   log(1 + x)   x    x *log(x + 1)
 | x*log(x + 1) dx = C + - - ---------- - -- + -------------
 |                       2       2        4          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
The answer [src]
1/4
$$\frac{1}{4}$$
=
=
1/4
$$\frac{1}{4}$$
1/4
Numerical answer [src]
0.25
0.25
The graph
Integral of x*ln(x+1) dx

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