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Solving integrals/ cosx/sqrt(2sinx+1)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 pi                    
 --                    
 2                     
  /                    
 |                     
 |       cos(x)        
 |  ---------------- dx
 |    ______________   
 |  \/ 2*sin(x) + 1    
 |                     
/                      
0
∫0π2cos⁡(x)2sin⁡(x)+1 dx\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

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

    Then let du=cos⁡(x)dx2sin⁡(x)+1du = \frac{\cos{\left(x \right)} dx}{\sqrt{2 \sin{\left(x \right)} + 1}} and substitute dudu:

    ∫1 du\int 1\, du

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 du=u\int 1\, du = u

    Now substitute uu back in:

    2sin⁡(x)+1\sqrt{2 \sin{\left(x \right)} + 1}

  2. Now simplify:

    2sin⁡(x)+1\sqrt{2 \sin{\left(x \right)} + 1}

  3. Add the constant of integration:

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

The answer is:

2sin⁡(x)+1+constant\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                           
 |                                           
/
∫cos⁡(x)2sin⁡(x)+1 dx=C+2sin⁡(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
0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.502
Plot the graph f(x)
The answer [src] 
       ___
-1 + \/ 3
−1+3-1 + \sqrt{3} 
=
= 
       ___
-1 + \/ 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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