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Derivative of:
1/3+2cosx-sinx
Identical expressions
one / three +2cosx-sinx
1 divide by 3 plus 2 co sinus of e of x minus sinus of x
one divide by three plus 2 co sinus of e of x minus sinus of x
1 divide by 3+2cosx-sinx
1/3+2cosx-sinxdx
Similar expressions
1/3-2cosx-sinx
1/(3+2*cos(x)-sin(x))
1/3+2cosx+sinx

Integral of 1/3+2cosx-sinx dx

The solution

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

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

Integral(1/3 + 2*cos(x) - sin(x), (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

1.55657760881727

1.55657760881727

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

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Identical expressions

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Integral of 1/3+2cosx-sinx dx

Limits of integration:

The graph:

Piecewise:

The solution