Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sqrt(x)/x
Integral of (-x)/(1+x^2)
Integral of (x^2-1)e^x
Integral of ln(x)*ln(x)
Identical expressions
(sin^ two)*x/ two
( sinus of squared ) multiply by x divide by 2
( sinus of to the power of two) multiply by x divide by two
(sin2)*x/2
sin2*x/2
(sin²)*x/2
(sin to the power of 2)*x/2
(sin^2)x/2
(sin2)x/2
sin2x/2
sin^2x/2
(sin^2)*x divide by 2
(sin^2)*x/2dx

Integral of (sin^2)*x/2 dx

The solution

You have entered [src]

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

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

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

Detail solution

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

$\int \frac{x \sin^{2}{\left(x \right)}}{2}\, dx = \frac{\int x \sin^{2}{\left(x \right)}\, dx}{2}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x^{2} \sin^{2}{\left(x \right)}}{4} + \frac{x^{2} \cos^{2}{\left(x \right)}}{4} - \frac{x \sin{\left(x \right)} \cos{\left(x \right)}}{2} - \frac{\cos^{2}{\left(x \right)}}{4}$
So, the result is: $\frac{x^{2} \sin^{2}{\left(x \right)}}{8} + \frac{x^{2} \cos^{2}{\left(x \right)}}{8} - \frac{x \sin{\left(x \right)} \cos{\left(x \right)}}{4} - \frac{\cos^{2}{\left(x \right)}}{8}$
Now simplify:

$\frac{x^{2}}{8} - \frac{x \sin{\left(2 x \right)}}{8} - \frac{\cos{\left(2 x \right)}}{16} - \frac{1}{16}$
Add the constant of integration:

$\frac{x^{2}}{8} - \frac{x \sin{\left(2 x \right)}}{8} - \frac{\cos{\left(2 x \right)}}{16} - \frac{1}{16}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{8} - \frac{x \sin{\left(2 x \right)}}{8} - \frac{\cos{\left(2 x \right)}}{16} - \frac{1}{16}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                      
 |                                                                       
 |    2                  2       2    2       2    2                     
 | sin (x)*x          cos (x)   x *cos (x)   x *sin (x)   x*cos(x)*sin(x)
 | --------- dx = C - ------- + ---------- + ---------- - ---------------
 |     2                 8          8            8               4       
 |                                                                       
/

$$\int \frac{x \sin^{2}{\left(x \right)}}{2}\, dx = C + \frac{x^{2} \sin^{2}{\left(x \right)}}{8} + \frac{x^{2} \cos^{2}{\left(x \right)}}{8} - \frac{x \sin{\left(x \right)} \cos{\left(x \right)}}{4} - \frac{\cos^{2}{\left(x \right)}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2                   
1   sin (1)   cos(1)*sin(1)
- + ------- - -------------
8      8            4

$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{4} + \frac{\sin^{2}{\left(1 \right)}}{8} + \frac{1}{8}$$

       2                   
1   sin (1)   cos(1)*sin(1)
- + ------- - -------------
8      8            4

$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{4} + \frac{\sin^{2}{\left(1 \right)}}{8} + \frac{1}{8}$$

1/8 + sin(1)^2/8 - cos(1)*sin(1)/4

Numerical answer [src]

0.0998469989309862

0.0998469989309862

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

Other calculators

How to use it?

Identical expressions

Integral of (sin^2)*x/2 dx

Limits of integration:

The graph:

Piecewise:

The solution