Integral of sinx-cos2x dx

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  1                       
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 |  (sin(x) - cos(2*x)) dx
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$$\int\limits_{0}^{1} \left(\sin{\left(x \right)} - \cos{\left(2 x \right)}\right)\, dx$$

Integral(sin(x) - cos(2*x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cos{\left(2 x \right)}\right)\, dx = - \int \cos{\left(2 x \right)}\, dx$
  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:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      1. 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: $- \frac{\sin{\left(2 x \right)}}{2}$
The result is: $- \frac{\sin{\left(2 x \right)}}{2} - \cos{\left(x \right)}$
Now simplify:

$- \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}$
Add the constant of integration:

$- \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                                       sin(2*x)
 | (sin(x) - cos(2*x)) dx = C - cos(x) - --------
 |                                          2    
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$$\int \left(\sin{\left(x \right)} - \cos{\left(2 x \right)}\right)\, dx = C - \frac{\sin{\left(2 x \right)}}{2} - \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

             sin(2)
1 - cos(1) - ------
               2

$$- \cos{\left(1 \right)} - \frac{\sin{\left(2 \right)}}{2} + 1$$

=

             sin(2)
1 - cos(1) - ------
               2

$$- \cos{\left(1 \right)} - \frac{\sin{\left(2 \right)}}{2} + 1$$

1 - cos(1) - sin(2)/2

Numerical answer [src]

0.00504898071901943

0.00504898071901943

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Integral of sinx-cos2x dx

Limits of integration:

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Piecewise:

The solution