Mister Exam

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Graphing y =:
2cos^2x
Identical expressions
2cos^2x
2 co sinus of e of squared x
2cos2x
2cos²x
2cos to the power of 2x
2cos^2xdx

Integral of 2cos^2x dx

The solution

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 157            
 ---            
 100            
  /             
 |              
 |       2      
 |  2*cos (x) dx
 |              
/               
0

$$\int\limits_{0}^{\frac{157}{100}} 2 \cos^{2}{\left(x \right)}\, dx$$

Integral(2*cos(x)^2, (x, 0, 157/100))

Detail solution

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

$\int 2 \cos^{2}{\left(x \right)}\, dx = 2 \int \cos^{2}{\left(x \right)}\, dx$
1. Rewrite the integrand:
  
  $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $\frac{\sin{\left(2 x \right)}}{4}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
So, the result is: $x + \frac{\sin{\left(2 x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

157      /157\    /157\
--- + cos|---|*sin|---|
100      \100/    \100/

$$\sin{\left(\frac{157}{100} \right)} \cos{\left(\frac{157}{100} \right)} + \frac{157}{100}$$

157      /157\    /157\
--- + cos|---|*sin|---|
100      \100/    \100/

$$\sin{\left(\frac{157}{100} \right)} \cos{\left(\frac{157}{100} \right)} + \frac{157}{100}$$

157/100 + cos(157/100)*sin(157/100)

Numerical answer [src]

1.57079632645824

1.57079632645824

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

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Identical expressions

Integral of 2cos^2x dx

Limits of integration:

The graph:

Piecewise:

The solution