Mister Exam

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Integral of d{x}:
Integral of 1/
Integral of abs(x)
Integral of xcos(x)dx
Integral of sin²(x)
Identical expressions
sqrt(two)sin(pix)
square root of (2) sinus of ( Pi x)
square root of (two) sinus of ( Pi x)
√(2)sin(pix)
sqrt2sinpix
sqrt(2)sin(pix)dx

Integral of sqrt(2)sin(pix) dx

The solution

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  /                   
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 |    ___             
 |  \/ 2 *sin(pi*x) dx
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$$\int\limits_{\frac{1}{3}}^{\frac{2}{3}} \sqrt{2} \sin{\left(\pi x \right)}\, dx$$

Integral(sqrt(2)*sin(pi*x), (x, 1/3, 2/3))

Detail solution

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

$\int \sqrt{2} \sin{\left(\pi x \right)}\, dx = \sqrt{2} \int \sin{\left(\pi x \right)}\, dx$
1. Let $u = \pi x$ .
  
  Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
  
  $\int \frac{\sin{\left(u \right)}}{\pi}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{\pi}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{\pi}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(\pi x \right)}}{\pi}$
So, the result is: $- \frac{\sqrt{2} \cos{\left(\pi x \right)}}{\pi}$
Add the constant of integration:

$- \frac{\sqrt{2} \cos{\left(\pi x \right)}}{\pi}+ \mathrm{constant}$

The answer is:

$- \frac{\sqrt{2} \cos{\left(\pi x \right)}}{\pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                            ___          
 |   ___                    \/ 2 *cos(pi*x)
 | \/ 2 *sin(pi*x) dx = C - ---------------
 |                                 pi      
/

$$\int \sqrt{2} \sin{\left(\pi x \right)}\, dx = C - \frac{\sqrt{2} \cos{\left(\pi x \right)}}{\pi}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___
\/ 2 
-----
  pi

$$\frac{\sqrt{2}}{\pi}$$

  ___
\/ 2 
-----
  pi

$$\frac{\sqrt{2}}{\pi}$$

sqrt(2)/pi

Numerical answer [src]

0.450158158078553

0.450158158078553

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

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Identical expressions

Integral of sqrt(2)sin(pix) dx

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Piecewise:

The solution