Integral of sin(3*x) dx

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

Integral(sin(3*x), (x, 0, pi/2))

Detail solution

Let $u = 3 x$ .

Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :

$\int \frac{\sin{\left(u \right)}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
Now substitute $u$ back in:

$- \frac{\cos{\left(3 x \right)}}{3}$
Add the constant of integration:

$- \frac{\cos{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

0.333333333333333

0.333333333333333

The graph

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