Mister Exam

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Integral of d{x}:
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Derivative of:
sin^2(5x)
Identical expressions
sin^ two (5x)
sinus of squared (5x)
sinus of to the power of two (5x)
sin2(5x)
sin25x
sin²(5x)
sin to the power of 2(5x)
sin^25x
sin^2(5x)dx
Similar expressions
sin^25x
(1/sin^2*5x-3/x^10)

Solving integrals/ sin^2(5x)

Integral of sin^2(5x) dx

The solution

You have entered [src]

  1             
  /             
 |              
 |     2        
 |  sin (5*x) dx
 |              
/               
0

$$\int\limits_{0}^{1} \sin^{2}{\left(5 x \right)}\, dx$$

Integral(sin(5*x)^2, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(5 x \right)} = \frac{1}{2} - \frac{\cos{\left(10 x \right)}}{2}$
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:
  
  $\int \left(- \frac{\cos{\left(10 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(10 x \right)}\, dx}{2}$
  1. Let $u = 10 x$ .
    
    Then let $du = 10 dx$ and substitute $\frac{du}{10}$ :
    
    $\int \frac{\cos{\left(u \right)}}{10}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{10}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{10}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(10 x \right)}}{10}$
  So, the result is: $- \frac{\sin{\left(10 x \right)}}{20}$
The result is: $\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}$
Add the constant of integration:

$\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                 
 |    2               x   sin(10*x)
 | sin (5*x) dx = C + - - ---------
 |                    2       20   
/

$$\int \sin^{2}{\left(5 x \right)}\, dx = C + \frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(5)*sin(5)
- - -------------
2         10

$$- \frac{\sin{\left(5 \right)} \cos{\left(5 \right)}}{10} + \frac{1}{2}$$

1   cos(5)*sin(5)
- - -------------
2         10

$$- \frac{\sin{\left(5 \right)} \cos{\left(5 \right)}}{10} + \frac{1}{2}$$

1/2 - cos(5)*sin(5)/10

Numerical answer [src]

0.527201055544468

0.527201055544468

The graph

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

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Identical expressions

Similar expressions

Integral of sin^2(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution