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Pi*(sinx)^2

Integral of Pi*(sin(x))^2 dx

The solution

You have entered [src]

 pi              
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 2               
  /              
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 |        2      
 |  pi*sin (x) dx
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0

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

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

Detail solution

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

$\int \pi \sin^{2}{\left(x \right)}\, dx = \pi \int \sin^{2}{\left(x \right)}\, dx$
1. Rewrite the integrand:
  
  $\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
  The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
So, the result is: $\pi \left(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}\right)$
Now simplify:

$\frac{\pi \left(2 x - \sin{\left(2 x \right)}\right)}{4}$
Add the constant of integration:

$\frac{\pi \left(2 x - \sin{\left(2 x \right)}\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{\pi \left(2 x - \sin{\left(2 x \right)}\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |       2                /x   sin(2*x)\
 | pi*sin (x) dx = C + pi*|- - --------|
 |                        \2      4    /
/

$$\int \pi \sin^{2}{\left(x \right)}\, dx = C + \pi \left(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2
pi 
---
 4

$$\frac{\pi^{2}}{4}$$

  2
pi 
---
 4

$$\frac{\pi^{2}}{4}$$

pi^2/4

Numerical answer [src]

2.46740110027234

2.46740110027234

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

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Integral of Pi*(sin(x))^2 dx

Limits of integration:

The graph:

Piecewise:

The solution