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integral of sin^5(t)cos^4(t)

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

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∫ sin⁵(t)cos⁴(t)dt

Solution

−cos⁵(t)5 +2cos⁷(t)7 −cos⁹(t)9 +C

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Solution steps

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∫ sin⁵(t)cos⁴(t)dt

Rewrite using trig identities

=∫ (1−cos²(t))²sin(t)cos⁴(t)dt

Apply u-substitution: ∫ −u⁴(1−u²)²du

=∫ −u⁴(1−u²)²du

Expand −u⁴(1−u²)²: −u⁴+2u⁶−u⁸

=∫ −u⁴+2u⁶−u⁸du

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

=−∫ u⁴du+∫ 2u⁶du−∫ u⁸du

∫ u⁴du=u⁵5

∫ 2u⁶du=2u⁷7

∫ u⁸du=u⁹9

=−u⁵5 +2u⁷7 −u⁹9

Substitute back u=cos(t)

=−cos⁵(t)5 +2cos⁷(t)7 −cos⁹(t)9

Add a constant to the solution

=−cos⁵(t)5 +2cos⁷(t)7 −cos⁹(t)9 +C

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

integral of sin^5(t)cos^4(t)

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