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integral of cos^3(t)

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∫ cos³(t)dt

Solution

sin(t)−sin³(t)3 +C

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∫ cos³(t)dt

Rewrite using trig identities

=∫ (1−sin²(t))cos(t)dt

Apply u-substitution: ∫ 1−u²du

=∫ 1−u²du

Apply the Sum Rule: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx

=∫ 1du−∫ u²du

∫ 1du=u

∫ u²du=u³3

=u−u³3

Substitute back u=sin(t)

=sin(t)−sin³(t)3

Add a constant to the solution

=sin(t)−sin³(t)3 +C

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

integral of cos^3(t)

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