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cos^6(x)

Integral of cos^6(x) dx

Limits of integration:

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The solution

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  1           
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 |     6      
 |  cos (x) dx
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$$\int\limits_{0}^{1} \cos^{6}{\left(x \right)}\, dx$$
Integral(cos(x)^6, (x, 0, 1))
Detail solution
  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. 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 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. 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:

            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. 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:

            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. 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. 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                                   
 |    6             sin (2*x)   sin(2*x)   3*sin(4*x)   5*x
 | cos (x) dx = C - --------- + -------- + ---------- + ---
 |                      48         4           64        16
/                                                          
$${{{{3\,\left({{\sin \left(4\,x\right)}\over{2}}+2\,x\right)}\over{ 16}}+{{\sin \left(2\,x\right)-{{\sin ^3\left(2\,x\right)}\over{3}} }\over{8}}+{{3\,\sin \left(2\,x\right)}\over{8}}+{{x}\over{4}} }\over{2}}$$
The graph
The answer [src]
        5                                    3          
5    cos (1)*sin(1)   5*cos(1)*sin(1)   5*cos (1)*sin(1)
-- + -------------- + --------------- + ----------------
16         6                 16                24       
$${{9\,\sin 4-4\,\sin ^32+48\,\sin 2+60}\over{192}}$$
=
=
        5                                    3          
5    cos (1)*sin(1)   5*cos(1)*sin(1)   5*cos (1)*sin(1)
-- + -------------- + --------------- + ----------------
16         6                 16                24       
$$\frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{6} + \frac{5 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{24} + \frac{5 \sin{\left(1 \right)} \cos{\left(1 \right)}}{16} + \frac{5}{16}$$
Numerical answer [src]
0.488686178391591
0.488686178391591
The graph
Integral of cos^6(x) dx

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