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integral of sin^8(x)cos^7(x)

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`x`²	`x`^□	log_□	√☐	^□√☐	≤	≥	□□	·	÷	`x`^◦	π
(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

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(☐)^′	∂∂`x`	∫_□^□	lim	∑	sin	cos	tan	cot	csc	sec

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∫ sin⁸(x)cos⁷(x)dx

Solution

sin⁹(x)9 −3sin¹¹(x)11 +3sin¹³(x)13 −sin¹⁵(x)15 +C

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∫ sin⁸(x)cos⁷(x)dx

Rewrite using trig identities

=∫ (1−sin²(x))³cos(x)sin⁸(x)dx

Apply u-substitution: ∫ u⁸(1−u²)³du

=∫ u⁸(1−u²)³du

Expand u⁸(1−u²)³: u⁸−3u¹⁰+3u¹²−u¹⁴

=∫ u⁸−3u¹⁰+3u¹²−u¹⁴du

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

=∫ u⁸du−∫ 3u¹⁰du+∫ 3u¹²du−∫ u¹⁴du

∫ u⁸du=u⁹9

∫ 3u¹⁰du=3u¹¹11

∫ 3u¹²du=3u¹³13

∫ u¹⁴du=u¹⁵15

=u⁹9 −3u¹¹11 +3u¹³13 −u¹⁵15

Substitute back u=sin(x)

=sin⁹(x)9 −3sin¹¹(x)11 +3sin¹³(x)13 −sin¹⁵(x)15

Add a constant to the solution

=sin⁹(x)9 −3sin¹¹(x)11 +3sin¹³(x)13 −sin¹⁵(x)15 +C

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

integral of sin^8(x)cos^7(x)

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