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Integral of x*sinx^2dx dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |       2        
 |  x*sin (x)*1 dx
 |                
/                 
0                 
$$\int\limits_{0}^{1} x \sin^{2}{\left(x \right)} 1\, dx$$
Integral(x*sin(x)^2*1, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

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

        1. Let .

          Then let and substitute :

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

            1. The integral of cosine is sine:

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Now evaluate the sub-integral.

  2. Integrate term-by-term:

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

      1. The integral of is when :

      So, the result is:

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

      1. There are multiple ways to do this integral.

        Method #1

        1. Let .

          Then let and substitute :

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

            1. The integral of sine is negative cosine:

            So, the result is:

          Now substitute back in:

        Method #2

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

          1. There are multiple ways to do this integral.

            Method #1

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            Method #2

            1. Let .

              Then let and substitute :

              1. The integral of is when :

              Now substitute back in:

          So, the result is:

      So, the result is:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                     
 |                       2                              
 |      2               x    cos(2*x)     /x   sin(2*x)\
 | x*sin (x)*1 dx = C - -- - -------- + x*|- - --------|
 |                      4       8         \2      4    /
/                                                       
$$-{{2\,x\,\sin \left(2\,x\right)+\cos \left(2\,x\right)-2\,x^2 }\over{8}}$$
The answer [src]
       2                   
1   sin (1)   cos(1)*sin(1)
- + ------- - -------------
4      4            2      
$${{1}\over{8}}-{{2\,\sin 2+\cos 2-2}\over{8}}$$
=
=
       2                   
1   sin (1)   cos(1)*sin(1)
- + ------- - -------------
4      4            2      
$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{4} + \frac{1}{4}$$
Numerical answer [src]
0.199693997861972
0.199693997861972

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