Integral of 1+cos2x dx

The solution

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

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

Integral(1 + cos(2*x), (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

    sin(2)
1 + ------
      2

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

    sin(2)
1 + ------
      2

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

1 + sin(2)/2

Numerical answer [src]

1.45464871341284

1.45464871341284

The graph

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

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Identical expressions

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Integral of 1+cos2x dx

Limits of integration:

The graph:

Piecewise:

The solution