Mister Exam

Other calculators

Solving integrals/ (x+2)/(x-3)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  6         
  /         
 |          
 |  x + 2   
 |  ----- dx
 |  x - 3   
 |          
/           
4
∫46x+2x−3 dx\int\limits_{4}^{6} \frac{x + 2}{x - 3}\, dx 
Integral((x + 2)/(x - 3), (x, 4, 6))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      x+2x−3=1+5x−3\frac{x + 2}{x - 3} = 1 + \frac{5}{x - 3}

    2. Integrate term-by-term:

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

        ∫1 dx=x\int 1\, dx = x

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

        ∫5x−3 dx=5∫1x−3 dx\int \frac{5}{x - 3}\, dx = 5 \int \frac{1}{x - 3}\, dx

        1. Let u=x−3u = x - 3.

          Then let du=dxdu = dx and substitute dudu:

          ∫1u du\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⁡(x−3)\log{\left(x - 3 \right)}

        So, the result is: 5log⁡(x−3)5 \log{\left(x - 3 \right)}

      The result is: x+5log⁡(x−3)x + 5 \log{\left(x - 3 \right)}

    Method #2

    1. Rewrite the integrand:

      x+2x−3=xx−3+2x−3\frac{x + 2}{x - 3} = \frac{x}{x - 3} + \frac{2}{x - 3}

    2. Integrate term-by-term:

      1. Rewrite the integrand:

        xx−3=1+3x−3\frac{x}{x - 3} = 1 + \frac{3}{x - 3}

      2. Integrate term-by-term:

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

          ∫1 dx=x\int 1\, dx = x

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

          ∫3x−3 dx=3∫1x−3 dx\int \frac{3}{x - 3}\, dx = 3 \int \frac{1}{x - 3}\, dx

          1. Let u=x−3u = x - 3.

            Then let du=dxdu = dx and substitute dudu:

            ∫1u du\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⁡(x−3)\log{\left(x - 3 \right)}

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

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

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

        ∫2x−3 dx=2∫1x−3 dx\int \frac{2}{x - 3}\, dx = 2 \int \frac{1}{x - 3}\, dx

        1. Let u=x−3u = x - 3.

          Then let du=dxdu = dx and substitute dudu:

          ∫1u du\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⁡(x−3)\log{\left(x - 3 \right)}

        So, the result is: 2log⁡(x−3)2 \log{\left(x - 3 \right)}

      The result is: x+3log⁡(x−3)+2log⁡(x−3)x + 3 \log{\left(x - 3 \right)} + 2 \log{\left(x - 3 \right)}

  2. Add the constant of integration:

    x+5log⁡(x−3)+constantx + 5 \log{\left(x - 3 \right)}+ \mathrm{constant}

The answer is:

x+5log⁡(x−3)+constantx + 5 \log{\left(x - 3 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 | x + 2                           
 | ----- dx = C + x + 5*log(-3 + x)
 | x - 3                           
 |                                 
/
∫x+2x−3 dx=C+x+5log⁡(x−3)\int \frac{x + 2}{x - 3}\, dx = C + x + 5 \log{\left(x - 3 \right)}
Simplify
The graph
4.06.04.24.44.64.85.05.25.45.65.8020
Plot the graph f(x)
The answer [src] 
2 + 5*log(3)
2+5log⁡(3)2 + 5 \log{\left(3 \right)} 
=
= 
2 + 5*log(3)
2+5log⁡(3)2 + 5 \log{\left(3 \right)} 
2 + 5*log(3)
Numerical answer [src] 
7.49306144334055
7.49306144334055

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

    © Mister Exam – Calculator