Integral of sin^2tdt dx

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 157            
 ---            
  50            
  /             
 |              
 |     2        
 |  sin (t)*1 dt
 |              
/               
0

$$\int\limits_{0}^{\frac{157}{50}} \sin^{2}{\left(t \right)} 1\, dt$$

Simplify

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(t \right)} 1 = \frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dt = \frac{t}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(2 t \right)}}{2}\right)\, dt = - \frac{\int \cos{\left(2 t \right)}\, dt}{2}$
  1. Let $u = 2 t$ .
    
    Then let $du = 2 dt$ 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:
    
    $\frac{\sin{\left(2 t \right)}}{2}$
  So, the result is: $- \frac{\sin{\left(2 t \right)}}{4}$
The result is: $\frac{t}{2} - \frac{\sin{\left(2 t \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                               
 |                                
 |    2               t   sin(2*t)
 | sin (t)*1 dt = C + - - --------
 |                    2      4    
/

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

Simplify

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The answer [src]

         /157\    /157\
      cos|---|*sin|---|
157      \ 50/    \ 50/
--- - -----------------
100           2

$$-{{25\,\sin \left({{157}\over{25}}\right)-157}\over{100}}$$

=

         /157\    /157\
      cos|---|*sin|---|
157      \ 50/    \ 50/
--- - -----------------
100           2

$$- \frac{\sin{\left(\frac{157}{50} \right)} \cos{\left(\frac{157}{50} \right)}}{2} + \frac{157}{100}$$

Simplify

Numerical answer [src]

1.57079632544828

1.57079632544828

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Identical expressions

Integral of sin^2tdt dx

Limits of integration:

The graph:

Piecewise:

The solution