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Identical expressions
(two sin5x+3cosx/2)
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(two sinus of 5x plus 3 co sinus of e of x divide by 2)
2sin5x+3cosx/2
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(2sin5x+3cosx/2)dx
Similar expressions
(2sin5x-3cosx/2)

Integral of (2sin5x+3cosx/2) dx

The solution

You have entered [src]

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

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

Integral(2*sin(5*x) + (3*cos(x))/2, (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:
  
  $\int 2 \sin{\left(5 x \right)}\, dx = 2 \int \sin{\left(5 x \right)}\, dx$
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      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:
    
    $- \frac{\cos{\left(5 x \right)}}{5}$
  So, the result is: $- \frac{2 \cos{\left(5 x \right)}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 \cos{\left(x \right)}}{2}\, dx = \frac{\int 3 \cos{\left(x \right)}\, dx}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \cos{\left(x \right)}\, dx = 3 \int \cos{\left(x \right)}\, dx$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    So, the result is: $3 \sin{\left(x \right)}$
  So, the result is: $\frac{3 \sin{\left(x \right)}}{2}$
The result is: $\frac{3 \sin{\left(x \right)}}{2} - \frac{2 \cos{\left(5 x \right)}}{5}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

1.54874160302655

1.54874160302655

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

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Identical expressions

Similar expressions

Integral of (2sin5x+3cosx/2) dx

Limits of integration:

The graph:

Piecewise:

The solution