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(sin^3)xcosxdx

Integral of (sin^3)xcosxdx dx

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The solution

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  1                      
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 |  sin (x)*x*cos(x)*1 dx
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$$\int\limits_{0}^{1} \sin^{3}{\left(x \right)} x \cos{\left(x \right)} 1\, dx$$
Integral(sin(x)^3*x*cos(x)*1, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      Method #2

      1. Rewrite the integrand:

      2. Let .

        Then let and substitute :

        1. 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 is the constant times the variable of integration:

          The result is:

        Now substitute back in:

    Now evaluate the sub-integral.

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

    1. Rewrite the integrand:

    2. There are multiple ways to do this integral.

      Method #1

      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. 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 cosine is sine:

                  So, the result is:

                Now substitute back in:

              So, the result is:

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

            The result is:

          So, the result is:

        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:

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

        The result is:

      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. 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 cosine is sine:

                  So, the result is:

                Now substitute back in:

              So, the result is:

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

            The result is:

          So, the result is:

        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:

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

        The result is:

    So, the result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                 
 |                                                              4   
 |    3                        3*x   sin(4*x)   sin(2*x)   x*sin (x)
 | sin (x)*x*cos(x)*1 dx = C - --- - -------- + -------- + ---------
 |                              32     128         16          4    
/                                                                   
$$-{{\sin \left(4\,x\right)-4\,x\,\cos \left(4\,x\right)-8\,\sin \left(2\,x\right)+16\,x\,\cos \left(2\,x\right)}\over{128}}$$
The graph
The answer [src]
       4           4           2       2           3                  3          
  3*cos (1)   5*sin (1)   3*cos (1)*sin (1)   3*cos (1)*sin(1)   5*sin (1)*cos(1)
- --------- + --------- - ----------------- + ---------------- + ----------------
      32          32              16                 32                 32       
$$-{{\sin 4-4\,\cos 4-8\,\sin 2+16\,\cos 2}\over{128}}$$
=
=
       4           4           2       2           3                  3          
  3*cos (1)   5*sin (1)   3*cos (1)*sin (1)   3*cos (1)*sin(1)   5*sin (1)*cos(1)
- --------- + --------- - ----------------- + ---------------- + ----------------
      32          32              16                 32                 32       
$$- \frac{3 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{16} - \frac{3 \cos^{4}{\left(1 \right)}}{32} + \frac{3 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{32} + \frac{5 \sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{32} + \frac{5 \sin^{4}{\left(1 \right)}}{32}$$
Numerical answer [src]
0.0943356000876032
0.0943356000876032
The graph
Integral of (sin^3)xcosxdx dx

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