integral of cos^5(x)

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(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

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∫ cos⁵(x)dx

Solution

sin(x)−2sin³(x)3 +sin⁵(x)5 +C

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∫ cos⁵(x)dx

Rewrite using trig identities

=∫ (1−sin²(x))²cos(x)dx

Apply u-substitution: ∫ (1−u²)²du

=∫ (1−u²)²du

Expand (1−u²)²: 1−2u²+u⁴

=∫ 1−2u²+u⁴du

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

=∫ 1du−∫ 2u²du+∫ u⁴du

∫ 1du=u

∫ 2u²du=2u³3

∫ u⁴du=u⁵5

=u−2u³3 +u⁵5

Substitute back u=sin(x)

=sin(x)−2sin³(x)3 +sin⁵(x)5

Add a constant to the solution

=sin(x)−2sin³(x)3 +sin⁵(x)5 +C

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

integral of cos^5(x)

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