Mister Exam

Other calculators

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

Integral of x/(1+x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

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

      ∫(−1x+1) dx=−∫1x+1 dx\int \left(- \frac{1}{x + 1}\right)\, dx = - \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: −log⁡(x+1)- \log{\left(x + 1 \right)}

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                             
 |                              
 |   x                          
 | ----- dx = C + x - log(1 + x)
 | 1 + x                        
 |                              
/
∫xx+1 dx=C+x−log⁡(x+1)\int \frac{x}{x + 1}\, dx = C + x - \log{\left(x + 1 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
1 - log(2)
1−log⁡(2)1 - \log{\left(2 \right)} 
=
= 
1 - log(2)
1−log⁡(2)1 - \log{\left(2 \right)} 
1 - log(2)
Numerical answer [src] 
0.306852819440055
0.306852819440055
The graph
Integral of x/(1+x) dx

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

    © Mister Exam – Calculator