Mister Exam

Other calculators

sinx^2dx
Solving integrals/ sinx^2dx

Integral of sinx^2dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

  1. Rewrite the integrand:

    sin⁡2(x)1=12−cos⁡(2x)2\sin^{2}{\left(x \right)} 1 = \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:

      ∫12 dx=x2\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:

      ∫(−cos⁡(2x)2) dx=−∫cos⁡(2x) dx2\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}

      1. Let u=2xu = 2 x.

        Then let du=2dxdu = 2 dx 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⁡(2x)2\frac{\sin{\left(2 x \right)}}{2}

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

    The result is: x2−sin⁡(2x)4\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}

  3. Add the constant of integration:

    x2−sin⁡(2x)4+constant\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}

The answer is:

x2−sin⁡(2x)4+constant\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
 |                                
 |    2               x   sin(2*x)
 | sin (x)*1 dx = C + - - --------
 |                    2      4    
/
x−sin⁡(2 x)22{{x-{{\sin \left(2\,x\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}
Simplify
Numerical answer [src] 
0.27267564329358
0.27267564329358
The graph
Integral of sinx^2dx dx

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

    © Mister Exam – Calculator