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Identical expressions
6cos(4x)-3sin(x)
6 co sinus of e of (4x) minus 3 sinus of (x)
6cos4x-3sinx
6cos(4x)-3sin(x)dx
Similar expressions
6cos(4x)+3sin(x)
6cos(4x)-3sinx

Integral of 6cos(4x)-3sin(x) dx

The solution

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  1                           
  /                           
 |                            
 |  (6*cos(4*x) - 3*sin(x)) dx
 |                            
/                             
0

$$\int\limits_{0}^{1} \left(- 3 \sin{\left(x \right)} + 6 \cos{\left(4 x \right)}\right)\, dx$$

Integral(6*cos(4*x) - 3*sin(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:
  
  $\int \left(- 3 \sin{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)}\, dx$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $3 \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 6 \cos{\left(4 x \right)}\, dx = 6 \int \cos{\left(4 x \right)}\, dx$
  1. Let $u = 4 x$ .
    
    Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(4 x \right)}}{4}$
  So, the result is: $\frac{3 \sin{\left(4 x \right)}}{2}$
The result is: $\frac{3 \sin{\left(4 x \right)}}{2} + 3 \cos{\left(x \right)}$
Add the constant of integration:

$\frac{3 \sin{\left(4 x \right)}}{2} + 3 \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{3 \sin{\left(4 x \right)}}{2} + 3 \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                      
 |                                             3*sin(4*x)
 | (6*cos(4*x) - 3*sin(x)) dx = C + 3*cos(x) + ----------
 |                                                 2     
/

$$\int \left(- 3 \sin{\left(x \right)} + 6 \cos{\left(4 x \right)}\right)\, dx = C + \frac{3 \sin{\left(4 x \right)}}{2} + 3 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                3*sin(4)
-3 + 3*cos(1) + --------
                   2

$$-3 + \frac{3 \sin{\left(4 \right)}}{2} + 3 \cos{\left(1 \right)}$$

                3*sin(4)
-3 + 3*cos(1) + --------
                   2

$$-3 + \frac{3 \sin{\left(4 \right)}}{2} + 3 \cos{\left(1 \right)}$$

-3 + 3*cos(1) + 3*sin(4)/2

Numerical answer [src]

-2.51429682535747

-2.51429682535747

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

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Identical expressions

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Integral of 6cos(4x)-3sin(x) dx

Limits of integration:

The graph:

Piecewise:

The solution