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Similar expressions
cosx^2+sin^2x

Integral of cosx^2-sin^2x dx

The solution

You have entered [src]

  p                       
  -                       
  4                       
  /                       
 |                        
 |  /   2         2   \   
 |  \cos (x) - sin (x)/ dx
 |                        
/                         
0

$$\int\limits_{0}^{\frac{p}{4}} \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)\, dx$$

Integral(cos(x)^2 - sin(x)^2, (x, 0, p/4))

Detail solution

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

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

Simplify

The answer [src]

   /p\    /p\
cos|-|*sin|-|
   \4/    \4/

$$\sin{\left(\frac{p}{4} \right)} \cos{\left(\frac{p}{4} \right)}$$

   /p\    /p\
cos|-|*sin|-|
   \4/    \4/

$$\sin{\left(\frac{p}{4} \right)} \cos{\left(\frac{p}{4} \right)}$$

cos(p/4)*sin(p/4)

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

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Integral of cosx^2-sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution