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Solving integrals/ 2*(1/sqrt(2)*cos(x)+1/sqrt(2)*sin(x))*(-1/sqrt(2)*sin(x))

Integral of 2(1/sqrt(2)cos(x)+1/sqrt(2)sin(x))(-1/sqrt(2)*sin(x)) dx

The solution

You have entered [src]

 2*pi                                   
   /                                    
  |                                     
  |    /cos(x)   sin(x)\  -1            
  |  2*|------ + ------|*-----*sin(x) dx
  |    |  ___      ___ |   ___          
  |    \\/ 2     \/ 2  / \/ 2           
  |                                     
 /                                      
 0

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

Integral((2*(cos(x)/sqrt(2) + sin(x)/sqrt(2)))*((-1/sqrt(2))*sin(x)), (x, 0, 2*pi))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $- \frac{1}{\sqrt{2}} \sin{\left(x \right)} 2 \left(\frac{\sin{\left(x \right)}}{\sqrt{2}} + \frac{\cos{\left(x \right)}}{\sqrt{2}}\right) = - \sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}$
2. 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:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      
      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. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{2}{\left(x \right)}}{2}$
    So, the result is: $\frac{\cos^{2}{\left(x \right)}}{2}$
  The result is: $- \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\cos^{2}{\left(x \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $- \frac{1}{\sqrt{2}} \sin{\left(x \right)} 2 \left(\frac{\sin{\left(x \right)}}{\sqrt{2}} + \frac{\cos{\left(x \right)}}{\sqrt{2}}\right) = - \sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}$
2. 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:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      
      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. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{2}{\left(x \right)}}{2}$
    So, the result is: $\frac{\cos^{2}{\left(x \right)}}{2}$
  The result is: $- \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\cos^{2}{\left(x \right)}}{2}$
Now simplify:

$- \frac{x}{2} + \frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)}}{4} + \frac{1}{4}$
Add the constant of integration:

$- \frac{x}{2} + \frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)}}{4} + \frac{1}{4}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2} + \frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)}}{4} + \frac{1}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                
 |                                              2                  
 |   /cos(x)   sin(x)\  -1                   cos (x)   x   sin(2*x)
 | 2*|------ + ------|*-----*sin(x) dx = C + ------- - - + --------
 |   |  ___      ___ |   ___                    2      2      4    
 |   \\/ 2     \/ 2  / \/ 2                                        
 |                                                                 
/

$$\int - \frac{1}{\sqrt{2}} \sin{\left(x \right)} 2 \left(\frac{\sin{\left(x \right)}}{\sqrt{2}} + \frac{\cos{\left(x \right)}}{\sqrt{2}}\right)\, dx = C - \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\cos^{2}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

            /    ___           \
        ___ |  \/ 2         ___|
      \/ 2 *|- ----- + pi*\/ 2 |
  1         \    2             /
- - - --------------------------
  2               2

$$- \frac{\sqrt{2} \left(- \frac{\sqrt{2}}{2} + \sqrt{2} \pi\right)}{2} - \frac{1}{2}$$

            /    ___           \
        ___ |  \/ 2         ___|
      \/ 2 *|- ----- + pi*\/ 2 |
  1         \    2             /
- - - --------------------------
  2               2

$$- \frac{\sqrt{2} \left(- \frac{\sqrt{2}}{2} + \sqrt{2} \pi\right)}{2} - \frac{1}{2}$$

-1/2 - sqrt(2)*(-sqrt(2)/2 + pi*sqrt(2))/2

Numerical answer [src]

-3.14159265358979

-3.14159265358979

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

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How to use it?

Identical expressions

Similar expressions

Integral of 2(1/sqrt(2)cos(x)+1/sqrt(2)sin(x))(-1/sqrt(2)*sin(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 2*(1/sqrt(2)*cos(x)+1/sqrt(2)*sin(x))*(-1/sqrt(2)*sin(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of 2(1/sqrt(2)cos(x)+1/sqrt(2)sin(x))(-1/sqrt(2)*sin(x)) dx