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sin^2x*cos^4xdx

Integral of sin^2x*cos^4xdx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1                     
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 |     2       4        
 |  sin (x)*cos (x)*1 dx
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$$\int\limits_{0}^{1} \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)} 1\, dx$$
Integral(sin(x)^2*cos(x)^4*1, (x, 0, 1))
Detail solution
  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 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. 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. The integral of cosine is sine:

          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:

    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. 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. 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 #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. 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. 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:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                    
 |                                               3     
 |    2       4               sin(4*x)   x    sin (2*x)
 | sin (x)*cos (x)*1 dx = C - -------- + -- + ---------
 |                               64      16       48   
/                                                      
$$\int \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)} 1\, dx = C + \frac{x}{16} + \frac{\sin^{3}{\left(2 x \right)}}{48} - \frac{\sin{\left(4 x \right)}}{64}$$
The graph
The answer [src]
        5                                3          
1    cos (1)*sin(1)   cos(1)*sin(1)   cos (1)*sin(1)
-- - -------------- + ------------- + --------------
16         6                16              24      
$$- \frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{6} + \frac{\sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{24} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{16} + \frac{1}{16}$$
=
=
        5                                3          
1    cos (1)*sin(1)   cos(1)*sin(1)   cos (1)*sin(1)
-- - -------------- + ------------- + --------------
16         6                16              24      
$$- \frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{6} + \frac{\sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{24} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{16} + \frac{1}{16}$$
Numerical answer [src]
0.0899881003364571
0.0899881003364571
The graph
Integral of sin^2x*cos^4xdx dx

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