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Integral of sin^4x*cos^6x dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1                   
  /                   
 |                    
 |     4       6      
 |  sin (x)*cos (x) dx
 |                    
/                     
0                     
$$\int\limits_{0}^{1} \sin^{4}{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$
Integral(sin(x)^4*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. 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. Let .

                Then let and substitute :

                1. The integral of is when :

                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. Let .

                Then let and substitute :

                1. The integral of is when :

                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:

          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. 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:

      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 #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 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 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 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 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. Rewrite the integrand:

        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. 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:

      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]
  /                                                              
 |                                        5                      
 |    4       6             sin(4*x)   sin (2*x)   sin(8*x)   3*x
 | sin (x)*cos (x) dx = C - -------- + --------- + -------- + ---
 |                            256         320        2048     256
/                                                                
$$\int \sin^{4}{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = C + \frac{3 x}{256} + \frac{\sin^{5}{\left(2 x \right)}}{320} - \frac{\sin{\left(4 x \right)}}{256} + \frac{\sin{\left(8 x \right)}}{2048}$$
The graph
The answer [src]
            7                9                3                5                            
 3    11*cos (1)*sin(1)   cos (1)*sin(1)   cos (1)*sin(1)   cos (1)*sin(1)   3*cos(1)*sin(1)
--- - ----------------- + -------------- + -------------- + -------------- + ---------------
256           80                10              128              160               256      
$$- \frac{11 \sin{\left(1 \right)} \cos^{7}{\left(1 \right)}}{80} + \frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{160} + \frac{\sin{\left(1 \right)} \cos^{9}{\left(1 \right)}}{10} + \frac{\sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{128} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{256} + \frac{3}{256}$$
=
=
            7                9                3                5                            
 3    11*cos (1)*sin(1)   cos (1)*sin(1)   cos (1)*sin(1)   cos (1)*sin(1)   3*cos(1)*sin(1)
--- - ----------------- + -------------- + -------------- + -------------- + ---------------
256           80                10              128              160               256      
$$- \frac{11 \sin{\left(1 \right)} \cos^{7}{\left(1 \right)}}{80} + \frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{160} + \frac{\sin{\left(1 \right)} \cos^{9}{\left(1 \right)}}{10} + \frac{\sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{128} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{256} + \frac{3}{256}$$
3/256 - 11*cos(1)^7*sin(1)/80 + cos(1)^9*sin(1)/10 + cos(1)^3*sin(1)/128 + cos(1)^5*sin(1)/160 + 3*cos(1)*sin(1)/256
Numerical answer [src]
0.0171006789396639
0.0171006789396639

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