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(sinx/ two +cosx/ four)
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( sinus of x divide by two plus co sinus of e of x divide by four)
sinx/2+cosx/4
(sinx divide by 2+cosx divide by 4)
(sinx/2+cosx/4)dx
Similar expressions
(sinx/2-cosx/4)

Integral of (sinx/2+cosx/4) dx

The solution

You have entered [src]

  n                     
  /                     
 |                      
 |  /sin(x)   cos(x)\   
 |  |------ + ------| dx
 |  \  2        4   /   
 |                      
/                       
0

$$\int\limits_{0}^{n} \left(\frac{\sin{\left(x \right)}}{2} + \frac{\cos{\left(x \right)}}{4}\right)\, dx$$

Integral(sin(x)/2 + cos(x)/4, (x, 0, n))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(x \right)}}{2}\, dx = \frac{\int \sin{\left(x \right)}\, dx}{2}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $- \frac{\cos{\left(x \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(x \right)}}{4}\, dx = \frac{\int \cos{\left(x \right)}\, dx}{4}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $\frac{\sin{\left(x \right)}}{4}$
The result is: $\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                                           
 | /sin(x)   cos(x)\          cos(x)   sin(x)
 | |------ + ------| dx = C - ------ + ------
 | \  2        4   /            2        4   
 |                                           
/

$$\int \left(\frac{\sin{\left(x \right)}}{2} + \frac{\cos{\left(x \right)}}{4}\right)\, dx = C + \frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{2}$$

Simplify

The answer [src]

1   cos(n)   sin(n)
- - ------ + ------
2     2        4

$$\frac{\sin{\left(n \right)}}{4} - \frac{\cos{\left(n \right)}}{2} + \frac{1}{2}$$

1   cos(n)   sin(n)
- - ------ + ------
2     2        4

$$\frac{\sin{\left(n \right)}}{4} - \frac{\cos{\left(n \right)}}{2} + \frac{1}{2}$$

1/2 - cos(n)/2 + sin(n)/4

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

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Integral of (sinx/2+cosx/4) dx

Limits of integration:

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Piecewise:

The solution