Mister Exam

Other calculators

sint^2
Solving integrals/ sint^2

Integral of sint^2 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1           
  /           
 |            
 |     2      
 |  sin (t) dt
 |            
/             
0
∫01sin⁡2(t) dt\int\limits_{0}^{1} \sin^{2}{\left(t \right)}\, dt 
Integral(sin(t)^2, (t, 0, 1))
Detail solution

  1. Rewrite the integrand:

    sin⁡2(t)=12−cos⁡(2t)2\sin^{2}{\left(t \right)} = \frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}

  2. Integrate term-by-term:

    1. The integral of a constant is the constant times the variable of integration:

      ∫12 dt=t2\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:

      ∫(−cos⁡(2t)2) dt=−∫cos⁡(2t) dt2\int \left(- \frac{\cos{\left(2 t \right)}}{2}\right)\, dt = - \frac{\int \cos{\left(2 t \right)}\, dt}{2}

      1. Let u=2tu = 2 t.

        Then let du=2dtdu = 2 dt and substitute du2\frac{du}{2}:

        ∫cos⁡(u)4 du\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:

          ∫cos⁡(u)2 du=∫cos⁡(u) du2\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}

          1. The integral of cosine is sine:

            ∫cos⁡(u) du=sin⁡(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

          So, the result is: sin⁡(u)2\frac{\sin{\left(u \right)}}{2}

        Now substitute uu back in:

        sin⁡(2t)2\frac{\sin{\left(2 t \right)}}{2}

      So, the result is: −sin⁡(2t)4- \frac{\sin{\left(2 t \right)}}{4}

    The result is: t2−sin⁡(2t)4\frac{t}{2} - \frac{\sin{\left(2 t \right)}}{4}

  3. Add the constant of integration:

    t2−sin⁡(2t)4+constant\frac{t}{2} - \frac{\sin{\left(2 t \right)}}{4}+ \mathrm{constant}

The answer is:

t2−sin⁡(2t)4+constant\frac{t}{2} - \frac{\sin{\left(2 t \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                             
 |                              
 |    2             t   sin(2*t)
 | sin (t) dt = C + - - --------
 |                  2      4    
/
t−sin⁡(2 t)22{{t-{{\sin \left(2\,t\right)}\over{2}}}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
1   cos(1)*sin(1)
- - -------------
2         2
−sin⁡2−24-{{\sin 2-2}\over{4}} 
=
= 
1   cos(1)*sin(1)
- - -------------
2         2
−sin⁡(1)cos⁡(1)2+12- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{1}{2}
Упростить
Numerical answer [src] 
0.27267564329358
0.27267564329358
The graph
Integral of sint^2 dx

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

    © Mister Exam – Calculator