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Integral of sin(5x)-cos(6x)
Identical expressions
sin(5x)-cos(6x)
sinus of (5x) minus co sinus of e of (6x)
sin5x-cos6x
sin(5x)-cos(6x)dx
Similar expressions
sin(5x)+cos(6x)
sin(5*x)-cos(6*x)

Integral of sin(5x)-cos(6x) dx

The solution

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  1                         
  /                         
 |                          
 |  (sin(5*x) - cos(6*x)) dx
 |                          
/                           
0

$$\int\limits_{0}^{1} \left(\sin{\left(5 x \right)} - \cos{\left(6 x \right)}\right)\, dx$$

Integral(sin(5*x) - cos(6*x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{25}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{5}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(5 x \right)}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cos{\left(6 x \right)}\right)\, dx = - \int \cos{\left(6 x \right)}\, dx$
  1. Let $u = 6 x$ .
    
    Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{\cos{\left(u \right)}}{36}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{6}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}$
      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:
    
    $\frac{\sin{\left(6 x \right)}}{6}$
  So, the result is: $- \frac{\sin{\left(6 x \right)}}{6}$
The result is: $- \frac{\sin{\left(6 x \right)}}{6} - \frac{\cos{\left(5 x \right)}}{5}$
Add the constant of integration:

$- \frac{\sin{\left(6 x \right)}}{6} - \frac{\cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer is:

$- \frac{\sin{\left(6 x \right)}}{6} - \frac{\cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                                cos(5*x)   sin(6*x)
 | (sin(5*x) - cos(6*x)) dx = C - -------- - --------
 |                                   5          6    
/

$$-{{\sin \left(6\,x\right)}\over{6}}-{{\cos \left(5\,x\right)}\over{ 5}}$$

Simplify

The answer [src]

1   cos(5)   sin(6)
- - ------ - ------
5     5        6

$$-{{5\,\sin 6+6\,\cos 5-6}\over{30}}$$

1   cos(5)   sin(6)
- - ------ - ------
5     5        6

$$- \frac{\cos{\left(5 \right)}}{5} - \frac{\sin{\left(6 \right)}}{6} + \frac{1}{5}$$

Упростить

Numerical answer [src]

0.189836812607176

0.189836812607176

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

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Integral of sin(5x)-cos(6x) dx

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The solution