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Integral of sin(x)-3cosx-x dx

The solution

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  p                           
  /                           
 |                            
 |  (sin(x) - 3*cos(x) - x) dx
 |                            
/                             
0

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

Integral(sin(x) - 3*cos(x) - x, (x, 0, p))

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. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \cos{\left(x \right)}\right)\, dx = - 3 \int \cos{\left(x \right)}\, dx$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    So, the result is: $- 3 \sin{\left(x \right)}$
  The result is: $- 3 \sin{\left(x \right)} - \cos{\left(x \right)}$
The result is: $- \frac{x^{2}}{2} - 3 \sin{\left(x \right)} - \cos{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

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

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

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

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

1 - cos(p) - 3*sin(p) - p^2/2

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

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Integral of sin(x)-3cosx-x dx

Limits of integration:

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Piecewise:

The solution