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Canonical form:
sin3x-cosx
Identical expressions
sin3x-cosx
sinus of 3x minus co sinus of e of x
sin3x-cosxdx
Similar expressions
sin3x+cosx

Solving integrals/ sin3x-cosx

Integral of sin3x-cosx dx

The solution

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

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

Integral(sin(3*x) - cos(x), (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1            cos(3)
- - sin(1) - ------
3              3

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

1            cos(3)
- - sin(1) - ------
3              3

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

1/3 - sin(1) - cos(3)/3

Numerical answer [src]

-0.178140152607748

-0.178140152607748

The graph

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

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Identical expressions

Similar expressions

Integral of sin3x-cosx dx

Limits of integration:

The graph:

Piecewise:

The solution