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cos12x-1

Solving integrals/ cos12x+1

Integral of cos12x+1 dx

The solution

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  1                   
  /                   
 |                    
 |  (cos(12*x) + 1) dx
 |                    
/                     
0

$$\int\limits_{0}^{1} \left(\cos{\left(12 x \right)} + 1\right)\, dx$$

Integral(cos(12*x) + 1, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Let $u = 12 x$ .
  
  Then let $du = 12 dx$ and substitute $\frac{du}{12}$ :
  
  $\int \frac{\cos{\left(u \right)}}{144}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{12}\, du = \frac{\int \cos{\left(u \right)}\, du}{12}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{12}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(12 x \right)}}{12}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x + \frac{\sin{\left(12 x \right)}}{12}$
Add the constant of integration:

$x + \frac{\sin{\left(12 x \right)}}{12}+ \mathrm{constant}$

The answer is:

$x + \frac{\sin{\left(12 x \right)}}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      
 |                              sin(12*x)
 | (cos(12*x) + 1) dx = C + x + ---------
 |                                  12   
/

$$\int \left(\cos{\left(12 x \right)} + 1\right)\, dx = C + x + \frac{\sin{\left(12 x \right)}}{12}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    sin(12)
1 + -------
       12

$$\frac{\sin{\left(12 \right)}}{12} + 1$$

    sin(12)
1 + -------
       12

$$\frac{\sin{\left(12 \right)}}{12} + 1$$

Упростить

Numerical answer [src]

0.95528559016663

0.95528559016663

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 cos12x+1 dx

Limits of integration:

The graph:

Piecewise:

The solution