Integral of cos3x+5dx dx

The solution

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  1                  
  /                  
 |                   
 |  (cos(3*x) + 5) dx
 |                   
/                    
0

$$\int\limits_{0}^{1} \left(\cos{\left(3 x \right)} + 5\right)\, dx$$

Integral(cos(3*x) + 5, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(3 x \right)}}{3}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 5\, dx = 5 x$
The result is: $5 x + \frac{\sin{\left(3 x \right)}}{3}$
Add the constant of integration:

$5 x + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$5 x + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      
 |                               sin(3*x)
 | (cos(3*x) + 5) dx = C + 5*x + --------
 |                                  3    
/

$$\int \left(\cos{\left(3 x \right)} + 5\right)\, dx = C + 5 x + \frac{\sin{\left(3 x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    sin(3)
5 + ------
      3

$$\frac{\sin{\left(3 \right)}}{3} + 5$$

    sin(3)
5 + ------
      3

$$\frac{\sin{\left(3 \right)}}{3} + 5$$

5 + sin(3)/3

Numerical answer [src]

5.04704000268662

5.04704000268662

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

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Integral of cos3x+5dx dx

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The solution