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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