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sin⁵x

Integral sin⁵x dx

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The solution

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  1           
  /           
 |            
 |     5      
 |  sin (x) dx
 |            
/             
0             
$$\int\limits_{0}^{1} \sin^{5}{\left(x \right)}\, dx$$
Integral(sin(x)^5, (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. 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:

      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. The integral of sine is negative cosine:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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:

      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. The integral of sine is negative cosine:

      The result is:

  3. Add the constant of integration:


The answer is:

The graph
The answer [src]
                 5           3   
8             cos (1)   2*cos (1)
-- - cos(1) - ------- + ---------
15               5          3    
$$- \cos{\left(1 \right)} - \frac{\cos^{5}{\left(1 \right)}}{5} + \frac{2 \cos^{3}{\left(1 \right)}}{3} + \frac{8}{15}$$
=
=
                 5           3   
8             cos (1)   2*cos (1)
-- - cos(1) - ------- + ---------
15               5          3    
$$- \cos{\left(1 \right)} - \frac{\cos^{5}{\left(1 \right)}}{5} + \frac{2 \cos^{3}{\left(1 \right)}}{3} + \frac{8}{15}$$
8/15 - cos(1) - cos(1)^5/5 + 2*cos(1)^3/3
Numerical answer [src]
0.0889743964515759
0.0889743964515759
The answer (Indefinite) [src]
  /                                             
 |                              5           3   
 |    5                      cos (x)   2*cos (x)
 | sin (x) dx = C - cos(x) - ------- + ---------
 |                              5          3    
/                                               
$$\int \sin^{5}{\left(x \right)}\, dx = C - \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$

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

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