Mister Exam

Other calculators

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                    
  /                    
 |                     
 |    ______________   
 |  \/ 2 + 2*cos(x)  dx
 |                     
/                      
0
∫012cos⁡(x)+2 dx\int\limits_{0}^{1} \sqrt{2 \cos{\left(x \right)} + 2}\, dx 
Integral(sqrt(2 + 2*cos(x)), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

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

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫2cos⁡(x)+1 dx=2∫cos⁡(x)+1 dx\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

      21−tan⁡2(x2)tan⁡2(x2)+1+1tan⁡2(x2)+1tan⁡(x2)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: 221−tan⁡2(x2)tan⁡2(x2)+1+1tan⁡2(x2)+1tan⁡(x2)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)}

  3. Now simplify:

    22cos⁡(x)+2tan⁡(x2)2 \sqrt{2 \cos{\left(x \right)} + 2} \tan{\left(\frac{x}{2} \right)}

  4. Add the constant of integration:

    22cos⁡(x)+2tan⁡(x2)+constant2 \sqrt{2 \cos{\left(x \right)} + 2} \tan{\left(\frac{x}{2} \right)}+ \mathrm{constant}

The answer is:

22cos⁡(x)+2tan⁡(x2)+constant2 \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(cos⁡x+1) sin⁡2x(cos⁡x+1)2+1{{4\,\sin x}\over{\left(\cos x+1\right)\,\sqrt{{{\sin ^2x}\over{ \left(\cos x+1\right)^2}}+1}}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9004
Plot the graph f(x)
The answer [src]
4 cos⁡21+2 cos⁡1+1 sin⁡1 sin⁡21+cos⁡21+2 cos⁡1+1(cos⁡1+1) sin⁡21+cos⁡31+3 cos⁡21+3 cos⁡1+1{{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
Integral of sqrt(2+2cos(x)) dx

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

    © Mister Exam – Calculator