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x^2cosx^3

Integral of x^2cosx^3 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 *cos (x) dx
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$$\int\limits_{0}^{1} x^{2} \cos^{3}{\left(x \right)}\, dx$$
Integral(x^2*cos(x)^3, (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 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:

      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 is when :

            Now substitute back in:

          So, the result is:

        1. The integral of cosine is sine:

        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 is when :

            Now substitute back in:

          So, the result is:

        1. The integral of cosine is sine:

        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. 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:

    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 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:

      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 cosine is sine:

      So, the result is:

    The result is:

  4. Now simplify:

  5. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
                                                                             /               3   \
  /                                                                          |            cos (x)|
 |                                      3         /     3            \   2*x*|-2*cos(x) - -------|
 |  2    3             14*sin(x)   2*sin (x)    2 |  sin (x)         |       \               3   /
 | x *cos (x) dx = C - --------- + --------- + x *|- ------- + sin(x)| - -------------------------
 |                         9           27         \     3            /               3            
/                                                                                                 
$${{\left(9\,x^2-2\right)\,\sin \left(3\,x\right)+6\,x\,\cos \left(3 \,x\right)+\left(81\,x^2-162\right)\,\sin x+162\,x\,\cos x}\over{108 }}$$
The graph
The answer [src]
        3            3           2                  2          
  22*sin (1)   14*cos (1)   5*cos (1)*sin(1)   4*sin (1)*cos(1)
- ---------- + ---------- - ---------------- + ----------------
      27           9               9                  3        
$${{7\,\sin 3+6\,\cos 3-81\,\sin 1+162\,\cos 1}\over{108}}$$
=
=
        3            3           2                  2          
  22*sin (1)   14*cos (1)   5*cos (1)*sin(1)   4*sin (1)*cos(1)
- ---------- + ---------- - ---------------- + ----------------
      27           9               9                  3        
$$- \frac{22 \sin^{3}{\left(1 \right)}}{27} - \frac{5 \sin{\left(1 \right)} \cos^{2}{\left(1 \right)}}{9} + \frac{14 \cos^{3}{\left(1 \right)}}{9} + \frac{4 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{3}$$
Numerical answer [src]
0.133497304240883
0.133497304240883
The graph
Integral of x^2cosx^3 dx

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