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Solving integrals/ 1/3*sin^2x*dx

Integral of 1/3sin^2xdx dx

The solution

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  1                 
  /                 
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 |         2        
 |  1/3*sin (x)*1 dx
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0

$$\int\limits_{0}^{1} \frac{1}{3} \sin^{2}{\left(x \right)} 1\, dx$$

Integral((1/3)*sin(x)^2*1, (x, 0, 1))

Detail solution

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

$\int \frac{1}{3} \sin^{2}{\left(x \right)} 1\, dx = \frac{\int \sin^{2}{\left(x \right)}\, dx}{3}$
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 $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        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: $\frac{x}{6} - \frac{\sin{\left(2 x \right)}}{12}$
Add the constant of integration:

$\frac{x}{6} - \frac{\sin{\left(2 x \right)}}{12}+ \mathrm{constant}$

The answer is:

$\frac{x}{6} - \frac{\sin{\left(2 x \right)}}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |        2               sin(2*x)   x
 | 1/3*sin (x)*1 dx = C - -------- + -
 |                           12      6
/

$${{x-{{\sin \left(2\,x\right)}\over{2}}}\over{6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(1)*sin(1)
- - -------------
6         6

$$-{{\sin 2-2}\over{12}}$$

1   cos(1)*sin(1)
- - -------------
6         6

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

Упростить

Numerical answer [src]

0.0908918810978599

0.0908918810978599

The graph

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

Other calculators

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Identical expressions

Integral of 1/3sin^2xdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of 1/3*sin^2x*dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 1/3sin^2xdx dx