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Solving integrals/ 2cos^2x-1

Integral of 2cos^2x-1 dx

The solution

You have entered [src]

 pi                   
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 4                    
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 |  /     2       \   
 |  \2*cos (x) - 1/ dx
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pi                    
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6

$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{4}} \left(2 \cos^{2}{\left(x \right)} - 1\right)\, dx$$

Simplify

Detail solution

Integrate term-by-term:
1. 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 $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{4}\, du$
        
        The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        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: $x + \frac{\sin{\left(2 x \right)}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \left(\left(-1\right) 1\right)\, dx = - x$
The result is: $\frac{\sin{\left(2 x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{\sin \left(2\,x\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ___
1   \/ 3 
- - -----
2     4

$${{\sin \left({{\pi}\over{2}}\right)-\sin \left({{\pi}\over{3}} \right)}\over{2}}$$

      ___
1   \/ 3 
- - -----
2     4

$$- \frac{\sqrt{3}}{4} + \frac{1}{2}$$

Simplify

Numerical answer [src]

0.0669872981077806

0.0669872981077806

The graph

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

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Identical expressions

Similar expressions

Integral of 2cos^2x-1 dx

Limits of integration:

The graph:

Piecewise:

The solution