Mister Exam

Other calculators


2x-3/(x-1)(x+2)

Integral of 2x-3/(x-1)(x+2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                         
  /                         
 |                          
 |  /        3          \   
 |  |2*x - -----*(x + 2)| dx
 |  \      x - 1        /   
 |                          
/                           
0                           
01(2x3x1(x+2))dx\int\limits_{0}^{1} \left(2 x - \frac{3}{x - 1} \left(x + 2\right)\right)\, dx
Integral(2*x - 3/(x - 1)*(x + 2), (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    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}

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

      (3x1(x+2))dx=3(x+2)x1dx\int \left(- \frac{3}{x - 1} \left(x + 2\right)\right)\, dx = - \int \frac{3 \left(x + 2\right)}{x - 1}\, dx

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

        3(x+2)x1dx=3x+2x1dx\int \frac{3 \left(x + 2\right)}{x - 1}\, dx = 3 \int \frac{x + 2}{x - 1}\, dx

        1. There are multiple ways to do this integral.

          Method #1

          1. Rewrite the integrand:

            x+2x1=1+3x1\frac{x + 2}{x - 1} = 1 + \frac{3}{x - 1}

          2. Integrate term-by-term:

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

              1dx=x\int 1\, dx = x

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

              3x1dx=31x1dx\int \frac{3}{x - 1}\, dx = 3 \int \frac{1}{x - 1}\, dx

              1. Let u=x1u = 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(x1)\log{\left(x - 1 \right)}

              So, the result is: 3log(x1)3 \log{\left(x - 1 \right)}

            The result is: x+3log(x1)x + 3 \log{\left(x - 1 \right)}

          Method #2

          1. Rewrite the integrand:

            x+2x1=xx1+2x1\frac{x + 2}{x - 1} = \frac{x}{x - 1} + \frac{2}{x - 1}

          2. Integrate term-by-term:

            1. Rewrite the integrand:

              xx1=1+1x1\frac{x}{x - 1} = 1 + \frac{1}{x - 1}

            2. Integrate term-by-term:

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

                1dx=x\int 1\, dx = x

              1. Let u=x1u = 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(x1)\log{\left(x - 1 \right)}

              The result is: x+log(x1)x + \log{\left(x - 1 \right)}

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

              2x1dx=21x1dx\int \frac{2}{x - 1}\, dx = 2 \int \frac{1}{x - 1}\, dx

              1. Let u=x1u = 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(x1)\log{\left(x - 1 \right)}

              So, the result is: 2log(x1)2 \log{\left(x - 1 \right)}

            The result is: x+log(x1)+2log(x1)x + \log{\left(x - 1 \right)} + 2 \log{\left(x - 1 \right)}

        So, the result is: 3x+9log(x1)3 x + 9 \log{\left(x - 1 \right)}

      So, the result is: 3x9log(x1)- 3 x - 9 \log{\left(x - 1 \right)}

    The result is: x23x9log(x1)x^{2} - 3 x - 9 \log{\left(x - 1 \right)}

  2. Add the constant of integration:

    x23x9log(x1)+constantx^{2} - 3 x - 9 \log{\left(x - 1 \right)}+ \mathrm{constant}


The answer is:

x23x9log(x1)+constantx^{2} - 3 x - 9 \log{\left(x - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                                       
 |                                                        
 | /        3          \           2                      
 | |2*x - -----*(x + 2)| dx = C + x  - 9*log(-1 + x) - 3*x
 | \      x - 1        /                                  
 |                                                        
/                                                         
(2x3x1(x+2))dx=C+x23x9log(x1)\int \left(2 x - \frac{3}{x - 1} \left(x + 2\right)\right)\, dx = C + x^{2} - 3 x - 9 \log{\left(x - 1 \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.900100000
The answer [src]
oo + 9*pi*I
+9iπ\infty + 9 i \pi
=
=
oo + 9*pi*I
+9iπ\infty + 9 i \pi
oo + 9*pi*i
Numerical answer [src]
394.818611075975
394.818611075975
The graph
Integral of 2x-3/(x-1)(x+2) dx

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