Mister Exam

Other calculators

Integral of (7-2x)*lg(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 1000                   
   /                    
  |                     
  |  (7 - 2*x)*log(x) dx
  |                     
 /                      
 100                    
$$\int\limits_{100}^{1000} \left(7 - 2 x\right) \log{\left(x \right)}\, dx$$
Integral((7 - 2*x)*log(x), (x, 100, 1000))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

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

          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:

          So, the result is:

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

          1. Use integration by parts:

            Let and let .

            Then .

            To find :

            1. The integral of the exponential function is itself.

            Now evaluate the sub-integral.

          2. The integral of the exponential function is itself.

          So, the result is:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        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:

        So, the result is:

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

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

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

          Now evaluate the sub-integral.

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

        So, the result is:

      The result is:

    Method #3

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. Integrate term-by-term:

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

        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:

      Now evaluate the sub-integral.

    2. Rewrite the integrand:

    3. Integrate term-by-term:

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

      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:

    Method #4

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        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:

        So, the result is:

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

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

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

          Now evaluate the sub-integral.

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

        So, the result is:

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                           2                               
 |                           x           2                    
 | (7 - 2*x)*log(x) dx = C + -- - 7*x - x *log(x) + 7*x*log(x)
 |                           2                                
/                                                             
$$\int \left(7 - 2 x\right) \log{\left(x \right)}\, dx = C - x^{2} \log{\left(x \right)} + \frac{x^{2}}{2} + 7 x \log{\left(x \right)} - 7 x$$
The graph
The answer [src]
488700 - 993000*log(1000) + 9300*log(100)
$$- 993000 \log{\left(1000 \right)} + 9300 \log{\left(100 \right)} + 488700$$
=
=
488700 - 993000*log(1000) + 9300*log(100)
$$- 993000 \log{\left(1000 \right)} + 9300 \log{\left(100 \right)} + 488700$$
488700 - 993000*log(1000) + 9300*log(100)
Numerical answer [src]
-6327872.90929957
-6327872.90929957

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