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Integral of (1-cos3x+2sin-x)dx dx

The solution

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                       2           
 |                                                       x    sin(3*x)
 | (1 - cos(3*x) + 2*sin(x) - x) dx = C + x - 2*cos(x) - -- - --------
 |                                                       2       3    
/

$$\int \left(- x + \left(\left(1 - \cos{\left(3 x \right)}\right) + 2 \sin{\left(x \right)}\right)\right)\, dx = C - \frac{x^{2}}{2} + x - \frac{\sin{\left(3 x \right)}}{3} - 2 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

1.3723553855771

1.3723553855771

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

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Integral of (1-cos3x+2sin-x)dx dx

Limits of integration:

The graph:

Piecewise:

The solution