Integral of sin6x dx

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  1            
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 |  sin(6*x) dx
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0

$$\int\limits_{0}^{1} \sin{\left(6 x \right)}\, dx$$

Integral(sin(6*x), (x, 0, 1))

Detail solution

Let $u = 6 x$ .

Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :

$\int \frac{\sin{\left(u \right)}}{36}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(u \right)}}{6}\, du = \frac{\int \sin{\left(u \right)}\, du}{6}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{6}$
Now substitute $u$ back in:

$- \frac{\cos{\left(6 x \right)}}{6}$
Add the constant of integration:

$- \frac{\cos{\left(6 x \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(6 x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          
 |                   cos(6*x)
 | sin(6*x) dx = C - --------
 |                      6    
/

$$-{{\cos \left(6\,x\right)}\over{6}}$$

Simplify

The answer [src]

1   cos(6)
- - ------
6     6

$${{1}\over{6}}-{{\cos 6}\over{6}}$$

1   cos(6)
- - ------
6     6

$$\frac{1}{6} - \frac{\cos{\left(6 \right)}}{6}$$

Упростить

Numerical answer [src]

0.00663828555827233

0.00663828555827233

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