any help with me understanding this problem would be very much appreciated.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

show,

[itex]^{π/2}_{0}[/itex][itex]\int[/itex] cos[itex]^{5}[/itex]xdx = 8/15

hence show

[itex]^{π/2}_{0}[/itex][itex]\int[/itex] sin[itex]^{5}[/itex]xdx = [itex]^{π/2}_{0}[/itex][itex]\int[/itex] cos[itex]^{5}[/itex]xdx

where,

cos[itex]^{5}[/itex]θ = [itex]\frac{cos5θ + 5cos3θ + 10cosθ}{16}[/itex]

sin[itex]^{5}[/itex]θ = [itex]\frac{sin5θ - 5sin3θ + 10sinθ}{16}[/itex]

2. Relevant equations

[itex]^{x}_{0}[/itex][itex]\int[/itex] cos(t)dt = [sin(t)][itex]^{x}_{0}[/itex]

= sin(x) - sin(0)

= sin(x)

3. The attempt at a solution

[itex]^{π/2}_{0}[/itex][itex]\int[/itex] cos[itex]^{5}[/itex]xdx = [itex]\frac{1}{16}[/itex] [itex]^{π/2}_{0}[/itex][itex]\int[/itex] (cos5θ + 5cos3θ + 10cosθ)dθ

= [itex]\frac{1}{16}[/itex] [sin5θ + 5sin3θ + 10sinθ][itex]^{π/2}_{0}[/itex]

= [itex]\frac{1}{16}[/itex] (1 + 5 + 10)

the answers say from the 2nd line of my attempt it should be...

= [itex]\frac{1}{16}[/itex] [[itex]\frac{sin5θ}{5}[/itex] + [itex]\frac{5sin3θ}{3}[/itex] + 10sinθ][itex]^{π/2}_{0}[/itex]

= [itex]\frac{1}{16}[/itex] ([itex]\frac{1}{5}[/itex] - [itex]\frac{5}{3}[/itex] + 10)

but i don't understand why the first term was divided by 5 and the second by 3,

or why the sign changed from plus to minus.

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# Trigonometric Applications - complex numbers

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