Mister Exam

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Integral of d{x}:
Integral of sqrt(2+2cos(x))
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Integral of sqrt(2+2sinx)
Integral of x⁴dx
Identical expressions
sqrt(two +2cos(x))
square root of (2 plus 2 co sinus of e of (x))
square root of (two plus 2 co sinus of e of (x))
√(2+2cos(x))
sqrt2+2cosx
sqrt(2+2cos(x))dx
Similar expressions
sqrt(2+2cosx)
(x+sinx)(sqrt(2+2*cosx))
sqrt(2-2cos(x))
sqrt(2+2cosx)

Solving integrals/ sqrt(2+2cos(x))

Integral of sqrt(2+2cos(x)) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\sqrt{2 \cos{\left(x \right)} + 2} = \sqrt{2} \sqrt{\cos{\left(x \right)} + 1}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \sqrt{2} \sqrt{\cos{\left(x \right)} + 1}\, dx = \sqrt{2} \int \sqrt{\cos{\left(x \right)} + 1}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $2 \sqrt{1 - \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \tan{\left(\frac{x}{2} \right)}$
So, the result is: $2 \sqrt{2} \sqrt{1 - \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \tan{\left(\frac{x}{2} \right)}$
Now simplify:

$2 \sqrt{2 \cos{\left(x \right)} + 2} \tan{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$2 \sqrt{2 \cos{\left(x \right)} + 2} \tan{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$2 \sqrt{2 \cos{\left(x \right)} + 2} \tan{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$${{4\,\sin x}\over{\left(\cos x+1\right)\,\sqrt{{{\sin ^2x}\over{ \left(\cos x+1\right)^2}}+1}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$${{4\,\sqrt{\cos ^21+2\,\cos 1+1}\,\sin 1\,\sqrt{\sin ^21+\cos ^21+2 \,\cos 1+1}}\over{\left(\cos 1+1\right)\,\sin ^21+\cos ^31+3\,\cos ^ 21+3\,\cos 1+1}}$$

(4*sqrt(cos(1)^2+2*cos(1)+1)*sin(1)*sqrt(sin(1)^2+cos(1)^2+2*cos(1)+1))/((cos(1)+1)*sin(1)^2+cos(1)^3+3*cos(1)^2+3*cos(1)+1)

Numerical answer [src]

1.91770215441681

1.91770215441681

The graph

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

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Identical expressions

Similar expressions

Integral of sqrt(2+2cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution