Integral of sinx² dx

The solution

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$$\int\limits_{0}^{1} \sin^{2}{\left(x \right)}\, dx$$

Integral(sin(x)^2, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
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 $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      1. 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: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                             
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 |    2             x   sin(2*x)
 | sin (x) dx = C + - - --------
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$${{x-{{\sin \left(2\,x\right)}\over{2}}}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(1)*sin(1)
- - -------------
2         2

$$-{{\sin 2-2}\over{4}}$$

1   cos(1)*sin(1)
- - -------------
2         2

$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{1}{2}$$

Simplify

Numerical answer [src]

0.27267564329358

0.27267564329358

The graph

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

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Integral of sinx² dx

Limits of integration:

The graph:

Piecewise:

The solution