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

Limits of integration:

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The graph:

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Piecewise:

The solution

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

    Method #1

    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. The integral of is when :

        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 is .

          Now substitute back in:

        So, the result is:

      The result is:

    Method #2

    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 is when :

          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:

        So, the result is:

      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 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 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                                     
 | 2*x  + 2*x - 3           2               
 | -------------- dx = C + x  - 3*log(1 + x)
 |     x + 1                                
 |                                          
/                                           
$$\int \frac{\left(2 x^{2} + 2 x\right) - 3}{x + 1}\, dx = C + x^{2} - 3 \log{\left(x + 1 \right)}$$
The graph
The answer [src]
1 - 3*log(2)
$$1 - 3 \log{\left(2 \right)}$$
=
=
1 - 3*log(2)
$$1 - 3 \log{\left(2 \right)}$$
1 - 3*log(2)
Numerical answer [src]
-1.07944154167984
-1.07944154167984

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