Mister Exam

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Identical expressions
one /(x+4sqrt(x))
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one divide by (x plus 4 square root of (x))
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1/x+4sqrtx
1 divide by (x+4sqrt(x))
1/(x+4sqrt(x))dx
Similar expressions
1/(x-4sqrt(x))

Integral of 1/(x+4sqrt(x)) dx

The solution

You have entered [src]

  9               
  /               
 |                
 |       1        
 |  ----------- dx
 |          ___   
 |  x + 4*\/ x    
 |                
/                 
1

$$\int\limits_{1}^{9} \frac{1}{4 \sqrt{x} + x}\, dx$$

Integral(1/(x + 4*sqrt(x)), (x, 1, 9))

Detail solution

Let $u = \sqrt{x}$ .

Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :

$\int \frac{2}{u + 4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u + 4}\, du = 2 \int \frac{1}{u + 4}\, du$
  1. Let $u = u + 4$ .
    
    Then let $du = du$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(u + 4 \right)}$
  So, the result is: $2 \log{\left(u + 4 \right)}$
Now substitute $u$ back in:

$2 \log{\left(\sqrt{x} + 4 \right)}$
Add the constant of integration:

$2 \log{\left(\sqrt{x} + 4 \right)}+ \mathrm{constant}$

The answer is:

$2 \log{\left(\sqrt{x} + 4 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |      1                    /      ___\
 | ----------- dx = C + 2*log\4 + \/ x /
 |         ___                          
 | x + 4*\/ x                           
 |                                      
/

$$\int \frac{1}{4 \sqrt{x} + x}\, dx = C + 2 \log{\left(\sqrt{x} + 4 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2*log(5) + 2*log(7)

$$- 2 \log{\left(5 \right)} + 2 \log{\left(7 \right)}$$

-2*log(5) + 2*log(7)

$$- 2 \log{\left(5 \right)} + 2 \log{\left(7 \right)}$$

-2*log(5) + 2*log(7)

Numerical answer [src]

0.672944473242426

0.672944473242426

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

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Identical expressions

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Integral of 1/(x+4sqrt(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution