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Integral of sin^10xcos^9x dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
 pi                    
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 |     10       9      
 |  sin  (x)*cos (x) dx
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$$\int\limits_{0}^{\frac{\pi}{2}} \sin^{10}{\left(x \right)} \cos^{9}{\left(x \right)}\, dx$$
Integral(sin(x)^10*cos(x)^9, (x, 0, pi/2))
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 is when :

        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 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 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 is when :

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      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. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      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. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                    
 |                                13           17         11         19           15   
 |    10       9             4*sin  (x)   4*sin  (x)   sin  (x)   sin  (x)   2*sin  (x)
 | sin  (x)*cos (x) dx = C - ---------- - ---------- + -------- + -------- + ----------
 |                               13           17          11         19          5     
/                                                                                      
$$\int \sin^{10}{\left(x \right)} \cos^{9}{\left(x \right)}\, dx = C + \frac{\sin^{19}{\left(x \right)}}{19} - \frac{4 \sin^{17}{\left(x \right)}}{17} + \frac{2 \sin^{15}{\left(x \right)}}{5} - \frac{4 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$$
The graph
The answer [src]
 128  
------
230945
$$\frac{128}{230945}$$
=
=
 128  
------
230945
$$\frac{128}{230945}$$
128/230945
Numerical answer [src]
0.000554244517092814
0.000554244517092814

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