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Integral of 2x/(✓x²+✓8)dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  4                  
  /                  
 |                   
 |       2*x         
 |  -------------- dx
 |       2           
 |    ___      ___   
 |  \/ x   + \/ 8    
 |                   
/                    
2                    
$$\int\limits_{2}^{4} \frac{2 x}{\left(\sqrt{x}\right)^{2} + \sqrt{8}}\, dx$$
Integral((2*x)/((sqrt(x))^2 + sqrt(8)), (x, 2, 4))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    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. 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. 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:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. 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:

    Method #3

    1. Rewrite the integrand:

    2. 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:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                      
 |                                                       
 |      2*x                          ___    /        ___\
 | -------------- dx = C + 2*x - 4*\/ 2 *log\x + 2*\/ 2 /
 |      2                                                
 |   ___      ___                                        
 | \/ x   + \/ 8                                         
 |                                                       
/                                                        
$$\int \frac{2 x}{\left(\sqrt{x}\right)^{2} + \sqrt{8}}\, dx = C + 2 x - 4 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$$
The graph
The answer [src]
        ___    /        ___\       ___    /        ___\
4 - 4*\/ 2 *log\4 + 2*\/ 2 / + 4*\/ 2 *log\2 + 2*\/ 2 /
$$- 4 \sqrt{2} \log{\left(2 \sqrt{2} + 4 \right)} + 4 + 4 \sqrt{2} \log{\left(2 + 2 \sqrt{2} \right)}$$
=
=
        ___    /        ___\       ___    /        ___\
4 - 4*\/ 2 *log\4 + 2*\/ 2 / + 4*\/ 2 *log\2 + 2*\/ 2 /
$$- 4 \sqrt{2} \log{\left(2 \sqrt{2} + 4 \right)} + 4 + 4 \sqrt{2} \log{\left(2 + 2 \sqrt{2} \right)}$$
4 - 4*sqrt(2)*log(4 + 2*sqrt(2)) + 4*sqrt(2)*log(2 + 2*sqrt(2))
Numerical answer [src]
2.03948371306291
2.03948371306291

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