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Integral of cosx/sqrt(2sinx+1) dx

The solution

You have entered [src]

 pi                    
 --                    
 2                     
  /                    
 |                     
 |       cos(x)        
 |  ---------------- dx
 |    ______________   
 |  \/ 2*sin(x) + 1    
 |                     
/                      
0

$$\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos{\left(x \right)}}{\sqrt{2 \sin{\left(x \right)} + 1}}\, dx$$

Integral(cos(x)/sqrt(2*sin(x) + 1), (x, 0, pi/2))

Detail solution

Let $u = \sqrt{2 \sin{\left(x \right)} + 1}$ .

Then let $du = \frac{\cos{\left(x \right)} dx}{\sqrt{2 \sin{\left(x \right)} + 1}}$ and substitute $du$ :

$\int 1\, du$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, du = u$
Now substitute $u$ back in:

$\sqrt{2 \sin{\left(x \right)} + 1}$
Now simplify:

$\sqrt{2 \sin{\left(x \right)} + 1}$
Add the constant of integration:

$\sqrt{2 \sin{\left(x \right)} + 1}+ \mathrm{constant}$

The answer is:

$\sqrt{2 \sin{\left(x \right)} + 1}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                                           
 |      cos(x)                 ______________
 | ---------------- dx = C + \/ 2*sin(x) + 1 
 |   ______________                          
 | \/ 2*sin(x) + 1                           
 |                                           
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

       ___
-1 + \/ 3

$$-1 + \sqrt{3}$$

       ___
-1 + \/ 3

$$-1 + \sqrt{3}$$

-1 + sqrt(3)

Numerical answer [src]

0.732050807568877

0.732050807568877

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

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Identical expressions

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Integral of cosx/sqrt(2sinx+1) dx

Limits of integration:

The graph:

Piecewise:

The solution