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Alik [6]
2 years ago
12

Evaluate the integral. (use c for the constant of integration.) 8 sin2(x) cos3(x) dx

Mathematics
1 answer:
Anastaziya [24]2 years ago
3 0
Important formula we will use.
First formula
2 cos α sin β = sin (α + β) - sin (α - β)
Second formula
∫ sin x dx = - cos x + c

We should change the trigonometry expression from multiplication into addition. Use the first formula above
∫ 8 sin (2x) cos (3x) dx
= ∫ 4 ( 2 sin (2x) cos (3x) dx)
= 4 ∫ 2 cos (3x) sin (2x) dx
= 4 ∫ sin (3x + 2x) - sin (3x - 2x) dx
= 4 ∫ sin 5x - sin x dx

After changing into addition, solve the integration. Use the second formula.
= 4 ∫ sin 5x - sin x dx
= 4(- \frac{1}{5} cos5x-cosx)+c
= - \frac{4}{5} cos5x-4cosx+c
This is the answer.
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