Mister Exam

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Graphing y =:
cos6x-sin3x
Derivative of:
cos6x-sin3x
Identical expressions
cos6x-sin3x
co sinus of e of 6x minus sinus of 3x
cos6x-sin3xdx
Similar expressions
cos6x+sin3x

Integral of cos6x-sin3x dx

The solution

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  1                         
  /                         
 |                          
 |  (cos(6*x) - sin(3*x)) dx
 |                          
/                           
0

$$\int\limits_{0}^{1} \left(- \sin{\left(3 x \right)} + \cos{\left(6 x \right)}\right)\, dx$$

Integral(cos(6*x) - sin(3*x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \sin{\left(3 x \right)}\right)\, dx = - \int \sin{\left(3 x \right)}\, dx$
  1. Let $u = 3 x$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\sin{\left(u \right)}}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
      1. The integral of sine is negative cosine:
        
        $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(3 x \right)}}{3}$
  So, the result is: $\frac{\cos{\left(3 x \right)}}{3}$
1. Let $u = 6 x$ .
  
  Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
  
  $\int \frac{\cos{\left(u \right)}}{6}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{6}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(6 x \right)}}{6}$
The result is: $\frac{\sin{\left(6 x \right)}}{6} + \frac{\cos{\left(3 x \right)}}{3}$
Now simplify:

$\frac{\left(\sin{\left(3 x \right)} + 1\right) \cos{\left(3 x \right)}}{3}$
Add the constant of integration:

$\frac{\left(\sin{\left(3 x \right)} + 1\right) \cos{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\left(\sin{\left(3 x \right)} + 1\right) \cos{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                                cos(3*x)   sin(6*x)
 | (cos(6*x) - sin(3*x)) dx = C + -------- + --------
 |                                   3          6    
/

$$\int \left(- \sin{\left(3 x \right)} + \cos{\left(6 x \right)}\right)\, dx = C + \frac{\sin{\left(6 x \right)}}{6} + \frac{\cos{\left(3 x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   cos(3)   sin(6)
- - + ------ + ------
  3     3        6

$$- \frac{1}{3} + \frac{\cos{\left(3 \right)}}{3} + \frac{\sin{\left(6 \right)}}{6}$$

  1   cos(3)   sin(6)
- - + ------ + ------
  3     3        6

$$- \frac{1}{3} + \frac{\cos{\left(3 \right)}}{3} + \frac{\sin{\left(6 \right)}}{6}$$

-1/3 + cos(3)/3 + sin(6)/6

Numerical answer [src]

-0.709900081899969

-0.709900081899969

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

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Identical expressions

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Integral of cos6x-sin3x dx

Limits of integration:

The graph:

Piecewise:

The solution