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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |  2*x + 3   
 |  ------- dx
 |  2*x + 1   
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{2 x + 3}{2 x + 1}\, dx$$
Integral((2*x + 3)/(2*x + 1), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

        1. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. 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. 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 is .

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Method #3

    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. Rewrite the integrand:

        2. 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. 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 is .

                So, the result is:

              Now substitute back in:

            So, the result is:

          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. 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 is .

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                 
 |                                  
 | 2*x + 3                          
 | ------- dx = C + x + log(1 + 2*x)
 | 2*x + 1                          
 |                                  
/                                   
$$\int \frac{2 x + 3}{2 x + 1}\, dx = C + x + \log{\left(2 x + 1 \right)}$$
The graph
The answer [src]
1 + log(3)
$$1 + \log{\left(3 \right)}$$
=
=
1 + log(3)
$$1 + \log{\left(3 \right)}$$
1 + log(3)
Numerical answer [src]
2.09861228866811
2.09861228866811

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