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Identical expressions
ctg(sqrtx+ five)/sqrtx
ctg( square root of x plus 5) divide by square root of x
ctg( square root of x plus five) divide by square root of x
ctg(√x+5)/√x
ctgsqrtx+5/sqrtx
ctg(sqrtx+5) divide by sqrtx
ctg(sqrtx+5)/sqrtxdx
Similar expressions
ctg(sqrtx-5)/sqrtx

Integral of ctg(sqrtx+5)/sqrtx dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |     /  ___    \   
 |  cot\\/ x  + 5/   
 |  -------------- dx
 |        ___        
 |      \/ x         
 |                   
/                    
0

$$\int\limits_{0}^{1} \frac{\cot{\left(\sqrt{x} + 5 \right)}}{\sqrt{x}}\, dx$$

Integral(cot(sqrt(x) + 5)/sqrt(x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x} + 5$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
  
  $\int 2 \cot{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cot{\left(u \right)}\, du = 2 \int \cot{\left(u \right)}\, du$
    1. Rewrite the integrand:
      
      $\cot{\left(u \right)} = \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}$
    2. Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\sin{\left(u \right)} \right)}$
    So, the result is: $2 \log{\left(\sin{\left(u \right)} \right)}$
  Now substitute $u$ back in:
  
  $2 \log{\left(\sin{\left(\sqrt{x} + 5 \right)} \right)}$
Method #2
1. Let $u = \sqrt{x}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
  
  $\int 2 \cot{\left(u + 5 \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cot{\left(u + 5 \right)}\, du = 2 \int \cot{\left(u + 5 \right)}\, du$
    1. Rewrite the integrand:
      
      $\cot{\left(u + 5 \right)} = \frac{\cos{\left(u + 5 \right)}}{\sin{\left(u + 5 \right)}}$
    2. Let $u = \sin{\left(u + 5 \right)}$ .
      
      Then let $du = \cos{\left(u + 5 \right)} du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\sin{\left(u + 5 \right)} \right)}$
    So, the result is: $2 \log{\left(\sin{\left(u + 5 \right)} \right)}$
  Now substitute $u$ back in:
  
  $2 \log{\left(\sin{\left(\sqrt{x} + 5 \right)} \right)}$
Now simplify:

$2 \log{\left(\sin{\left(\sqrt{x} + 5 \right)} \right)}$
Add the constant of integration:

$2 \log{\left(\sin{\left(\sqrt{x} + 5 \right)} \right)}+ \mathrm{constant}$

The answer is:

$2 \log{\left(\sin{\left(\sqrt{x} + 5 \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                              
 |    /  ___    \                               
 | cot\\/ x  + 5/               /   /  ___    \\
 | -------------- dx = C + 2*log\sin\\/ x  + 5//
 |       ___                                    
 |     \/ x                                     
 |                                              
/

$$\int \frac{\cot{\left(\sqrt{x} + 5 \right)}}{\sqrt{x}}\, dx = C + 2 \log{\left(\sin{\left(\sqrt{x} + 5 \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

-2.46622438799234

-2.46622438799234

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

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Identical expressions

Similar expressions

Integral of ctg(sqrtx+5)/sqrtx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2