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Integral of xlog(x)-x+c1x+c2 dx

Limits of integration:

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Piecewise:

The solution

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  1                              
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 |  (x*log(x) - x + c1*x + c2) dx
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0                                
$$\int\limits_{0}^{1} \left(c_{2} + \left(c_{1} x + \left(x \log{\left(x \right)} - x\right)\right)\right)\, dx$$
Integral(x*log(x) - x + c1*x + c2, (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

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

    1. Integrate term-by-term:

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

        1. The integral of is when :

        So, the result is:

      1. Integrate term-by-term:

        1. There are multiple ways to do this integral.

          Method #1

          1. Let .

            Then let and substitute :

            1. Use integration by parts:

              Let and let .

              Then .

              To find :

              1. Let .

                Then let and substitute :

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

                  1. The integral of the exponential function is itself.

                  So, the result is:

                Now substitute back in:

              Now evaluate the sub-integral.

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

              1. Let .

                Then let and substitute :

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

                  1. The integral of the exponential function is itself.

                  So, the result is:

                Now substitute back in:

              So, the result is:

            Now substitute back in:

          Method #2

          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. The integral of is when :

            So, the result is:

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

          1. The integral of is when :

          So, the result is:

        The result is:

      The result is:

    The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       2              2    2       
 |                                     3*x           c1*x    x *log(x)
 | (x*log(x) - x + c1*x + c2) dx = C - ---- + c2*x + ----- + ---------
 |                                      4              2         2    
/                                                                     
$$\int \left(c_{2} + \left(c_{1} x + \left(x \log{\left(x \right)} - x\right)\right)\right)\, dx = C + \frac{c_{1} x^{2}}{2} + c_{2} x + \frac{x^{2} \log{\left(x \right)}}{2} - \frac{3 x^{2}}{4}$$
The answer [src]
  3        c1
- - + c2 + --
  4        2 
$$\frac{c_{1}}{2} + c_{2} - \frac{3}{4}$$
=
=
  3        c1
- - + c2 + --
  4        2 
$$\frac{c_{1}}{2} + c_{2} - \frac{3}{4}$$
-3/4 + c2 + c1/2

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