Integral of 4cos3x dx

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 pi              
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 4               
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 |  4*cos(3*x) dx
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0

$$\int\limits_{0}^{\frac{\pi}{4}} 4 \cos{\left(3 x \right)}\, dx$$

Simplify

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 4 \cos{\left(3 x \right)}\, dx = 4 \int \cos{\left(3 x \right)}\, dx$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{9}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{3}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
    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:
  
  $\frac{\sin{\left(3 x \right)}}{3}$
So, the result is: $\frac{4 \sin{\left(3 x \right)}}{3}$
Add the constant of integration:

$\frac{4 \sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{4 \sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                     4*sin(3*x)
 | 4*cos(3*x) dx = C + ----------
 |                         3     
/

$${{4\,\sin \left(3\,x\right)}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___
2*\/ 2 
-------
   3

$$\frac{2 \sqrt{2}}{3}$$

    ___
2*\/ 2 
-------
   3

$$\frac{2 \sqrt{2}}{3}$$

Simplify

Numerical answer [src]

0.942809041582063

0.942809041582063

The graph

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