Integral of 2cos(3x) dx

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  x              
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 |  2*cos(3*x) dx
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\int\limits_{0}^{\frac{x}{2}} 2 \cos{\left(3 x \right)}\, dx

Integral(2*cos(3*x), (x, 0, x/2))

Detail solution

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

$\int 2 \cos{\left(3 x \right)}\, dx = 2 \int \cos{\left(3 x \right)}\, dx$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{3}\, du$
  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:
  
  $\frac{\sin{\left(3 x \right)}}{3}$
So, the result is: $\frac{2 \sin{\left(3 x \right)}}{3}$
Add the constant of integration:

$\frac{2 \sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

\frac{2 \sin{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                              
 |                     2*sin(3*x)
 | 2*cos(3*x) dx = C + ----------
 |                         3     
/

\int 2 \cos{\left(3 x \right)}\, dx = C + \frac{2 \sin{\left(3 x \right)}}{3}

Simplify

The answer [src]

     /3*x\
2*sin|---|
     \ 2 /
----------
    3

\frac{2 \sin{\left(\frac{3 x}{2} \right)}}{3}

     /3*x\
2*sin|---|
     \ 2 /
----------
    3

\frac{2 \sin{\left(\frac{3 x}{2} \right)}}{3}

2*sin(3*x/2)/3

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

Integral of 2*cos(3*x) dx