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Integral of d{x}:
Integral of 6
Integral of x^2*e^(3*x)
Integral of cos²(x)
Integral of y^2
Graphing y =:
sin((x-pi)/6)
Identical expressions
sin((x-pi)/ six)
sinus of ((x minus Pi ) divide by 6)
sinus of ((x minus Pi ) divide by six)
sinx-pi/6
sin((x-pi) divide by 6)
sin((x-pi)/6)dx
Similar expressions
sin((x+pi)/6)
sin(x-pi/6)^2

Integral of sin((x-pi)/6) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |     /x - pi\   
 |  sin|------| dx
 |     \  6   /   
 |                
/                 
0

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

Integral(sin((x - pi)/6), (x, 0, 1))

Detail solution

Let $u = \frac{x - \pi}{6}$ .

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

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

$- 6 \cos{\left(\frac{x - \pi}{6} \right)}$
Now simplify:

$- 6 \sin{\left(\frac{x}{6} + \frac{\pi}{3} \right)}$
Add the constant of integration:

$- 6 \sin{\left(\frac{x}{6} + \frac{\pi}{3} \right)}+ \mathrm{constant}$

The answer is:

$- 6 \sin{\left(\frac{x}{6} + \frac{\pi}{3} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                   
 |    /x - pi\               /x - pi\
 | sin|------| dx = C - 6*cos|------|
 |    \  6   /               \  6   /
 |                                   
/

$$\int \sin{\left(\frac{x - \pi}{6} \right)}\, dx = C - 6 \cos{\left(\frac{x - \pi}{6} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       /1   pi\       ___
- 6*sin|- + --| + 3*\/ 3 
       \6   3 /

$$- 6 \sin{\left(\frac{1}{6} + \frac{\pi}{3} \right)} + 3 \sqrt{3}$$

       /1   pi\       ___
- 6*sin|- + --| + 3*\/ 3 
       \6   3 /

$$- 6 \sin{\left(\frac{1}{6} + \frac{\pi}{3} \right)} + 3 \sqrt{3}$$

-6*sin(1/6 + pi/3) + 3*sqrt(3)

Numerical answer [src]

-0.425686517195053

-0.425686517195053

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

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Integral of sin((x-pi)/6) dx

Limits of integration:

The graph:

Piecewise:

The solution