Mister Exam

Other calculators

(x+3)/(x-1)
Solving integrals/ (x+3)/(x-1)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1         
  /         
 |          
 |  x + 3   
 |  ----- dx
 |  x - 1   
 |          
/           
0
∫01x+3x−1 dx\int\limits_{0}^{1} \frac{x + 3}{x - 1}\, dx 
Integral((x + 3)/(x - 1), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      x+3x−1=1+4x−1\frac{x + 3}{x - 1} = 1 + \frac{4}{x - 1}

    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:

        ∫4x−1 dx=4∫1x−1 dx\int \frac{4}{x - 1}\, dx = 4 \int \frac{1}{x - 1}\, dx

        1. Let u=x−1u = x - 1.

          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−1)\log{\left(x - 1 \right)}

        So, the result is: 4log⁡(x−1)4 \log{\left(x - 1 \right)}

      The result is: x+4log⁡(x−1)x + 4 \log{\left(x - 1 \right)}

    Method #2

    1. Rewrite the integrand:

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

    2. Integrate term-by-term:

      1. Rewrite the integrand:

        xx−1=1+1x−1\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:

          ∫1 dx=x\int 1\, dx = x

        1. Let u=x−1u = x - 1.

          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−1)\log{\left(x - 1 \right)}

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

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

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

        1. Let u=x−1u = x - 1.

          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−1)\log{\left(x - 1 \right)}

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

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

  2. Add the constant of integration:

    x+4log⁡(x−1)+constantx + 4 \log{\left(x - 1 \right)}+ \mathrm{constant}

The answer is:

x+4log⁡(x−1)+constantx + 4 \log{\left(x - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 | x + 3                           
 | ----- dx = C + x + 4*log(-1 + x)
 | x - 1                           
 |                                 
/
∫x+3x−1 dx=C+x+4log⁡(x−1)\int \frac{x + 3}{x - 1}\, dx = C + x + 4 \log{\left(x - 1 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-5000050000
Plot the graph f(x)
The answer [src] 
-oo - 4*pi*I
−∞−4iπ-\infty - 4 i \pi 
=
= 
-oo - 4*pi*I
−∞−4iπ-\infty - 4 i \pi 
-oo - 4*pi*i
Numerical answer [src] 
-175.363827144878
-175.363827144878
The graph
Integral of (x+3)/(x-1) dx

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

    © Mister Exam – Calculator