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x^2sinx^3dx

Integral of x^2sinx^3dx dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1                
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 |   2    3        
 |  x *sin (x)*1 dx
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0                  
$$\int\limits_{0}^{1} x^{2} \sin^{3}{\left(x \right)} 1\, dx$$
Integral(x^2*sin(x)^3*1, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Rewrite the integrand:

    2. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          1. The integral of is when :

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

          The result is:

        Now substitute back in:

      Method #2

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        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 is when :

              So, the result is:

            Now substitute back in:

          So, the result is:

        1. The integral of sine is negative cosine:

        The result is:

      Method #3

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        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 is when :

              So, the result is:

            Now substitute back in:

          So, the result is:

        1. The integral of sine is negative cosine:

        The result is:

    Now evaluate the sub-integral.

  2. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Integrate term-by-term:

      1. Rewrite the integrand:

      2. Let .

        Then let and substitute :

        1. 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. The integral of is when :

            So, the result is:

          The result is:

        Now substitute back in:

      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:

      The result is:

    Now evaluate the sub-integral.

  3. Integrate term-by-term:

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

      1. Rewrite the integrand:

      2. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          1. The integral of is when :

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

          The result is:

        Now substitute back in:

      So, the result is:

    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:

    The result is:

  4. Now simplify:

  5. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
                                                                              /               3   \
  /                                                                           |            sin (x)|
 |                            3                     /             3   \   2*x*|-2*sin(x) - -------|
 |  2    3               2*cos (x)   14*cos(x)    2 |          cos (x)|       \               3   /
 | x *sin (x)*1 dx = C - --------- + --------- + x *|-cos(x) + -------| - -------------------------
 |                           27          9          \             3   /               3            
/                                                                                                  
$$-{{6\,x\,\sin \left(3\,x\right)+\left(2-9\,x^2\right)\,\cos \left(3 \,x\right)-162\,x\,\sin x+\left(81\,x^2-162\right)\,\cos x}\over{108 }}$$
The graph
The answer [src]
             3            3           2                  2          
  40   14*sin (1)   22*cos (1)   4*cos (1)*sin(1)   5*sin (1)*cos(1)
- -- + ---------- + ---------- + ---------------- + ----------------
  27       9            27              3                  9        
$$-{{6\,\sin 3-7\,\cos 3-162\,\sin 1-81\,\cos 1}\over{108}}-{{40 }\over{27}}$$
=
=
             3            3           2                  2          
  40   14*sin (1)   22*cos (1)   4*cos (1)*sin(1)   5*sin (1)*cos(1)
- -- + ---------- + ---------- + ---------------- + ----------------
  27       9            27              3                  9        
$$- \frac{40}{27} + \frac{22 \cos^{3}{\left(1 \right)}}{27} + \frac{5 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{9} + \frac{4 \sin{\left(1 \right)} \cos^{2}{\left(1 \right)}}{3} + \frac{14 \sin^{3}{\left(1 \right)}}{9}$$
Numerical answer [src]
0.113945544348484
0.113945544348484
The graph
Integral of x^2sinx^3dx dx

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