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Identical expressions
(5cos(2x)+3sin(x))
(5 co sinus of e of (2x) plus 3 sinus of (x))
5cos2x+3sinx
(5cos(2x)+3sin(x))dx
Similar expressions
(5cos(2x)-3sin(x))
(5cos(2x)+3sinx)

Integral of (5cos(2x)+3sin(x)) dx

The solution

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  1                           
  /                           
 |                            
 |  (5*cos(2*x) + 3*sin(x)) dx
 |                            
/                             
0

$$\int\limits_{0}^{1} \left(3 \sin{\left(x \right)} + 5 \cos{\left(2 x \right)}\right)\, dx$$

Integral(5*cos(2*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:
  
  $\int 3 \sin{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)}\, dx$
  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:
  
  $\int 5 \cos{\left(2 x \right)}\, dx = 5 \int \cos{\left(2 x \right)}\, dx$
  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: $\frac{5 \sin{\left(2 x \right)}}{2}$
The result is: $\frac{5 \sin{\left(2 x \right)}}{2} - 3 \cos{\left(x \right)}$
Now simplify:

$\left(5 \sin{\left(x \right)} - 3\right) \cos{\left(x \right)}$
Add the constant of integration:

$\left(5 \sin{\left(x \right)} - 3\right) \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\left(5 \sin{\left(x \right)} - 3\right) \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                      
 |                                             5*sin(2*x)
 | (5*cos(2*x) + 3*sin(x)) dx = C - 3*cos(x) + ----------
 |                                                 2     
/

$$\int \left(3 \sin{\left(x \right)} + 5 \cos{\left(2 x \right)}\right)\, dx = C + \frac{5 \sin{\left(2 x \right)}}{2} - 3 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$$- 3 \cos{\left(1 \right)} + \frac{5 \sin{\left(2 \right)}}{2} + 3$$

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

$$- 3 \cos{\left(1 \right)} + \frac{5 \sin{\left(2 \right)}}{2} + 3$$

3 - 3*cos(1) + 5*sin(2)/2

Numerical answer [src]

3.65233664945979

3.65233664945979

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

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Identical expressions

Similar expressions

Integral of (5cos(2x)+3sin(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution