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Integral of sinxdx/2+cos^2(x) dx

The solution

You have entered [src]

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

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

Integral(sin(x)/2 + cos(x)^2, (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:
  
  $\int \frac{\sin{\left(x \right)}}{2}\, dx = \frac{\int \sin{\left(x \right)}\, dx}{2}$
  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. Rewrite the integrand:
  
  $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $\frac{\sin{\left(2 x \right)}}{4}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} - \frac{\cos{\left(x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.957173203772351

0.957173203772351

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

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Identical expressions

Similar expressions

Integral of sinxdx/2+cos^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution