## Saturday, December 25, 2021

### 12 Days of Christmas Integrals: ∫ π ∙ cos(x) ∙ sin(x) dx

12 Days of Christmas Integrals:  ∫ π ∙ cos(x) ∙ sin(x) dx

In the spirit of the Christmas Holiday, I am presenting you with the 12 Days of Christmas Integrals!

On the First day of Christmas Integrals, the integral featured today is...

∫ π ∙ cos(x) ∙ sin(x) dx

Let's start off with using the trigonometric identity

sin(2 ∙ x) = 2 ∙ cos(x) ∙ sin(x)

1/2 ∙ sin(2 ∙x) = cos(x) ∙ sin(x)

Then:

∫ π ∙ cos(x) ∙ sin(x) dx

= π ∙ ∫ 1/2 ∙ sin(2 ∙ x) dx

= π/2 ∙ ∫ sin(2 ∙ x) dx

Multiply by both 1/2 and 2:

= π/4 ∙ ∫ 2 ∙ sin(2 ∙ x) dx

= -π/4 ∙ cos(2 ∙ x) + C

[ ∫ sin x dx = -cos x + C,  angles are in radians ]

Eddie

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