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integral of sin^7(x)cos^6(x)

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∫ sin⁷(x)cos⁶(x)dx

Solution

−cos⁷(x)7 +cos⁹(x)3 −3cos¹¹(x)11 +cos¹³(x)13 +C

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∫ sin⁷(x)cos⁶(x)dx

Rewrite using trig identities

=∫ (1−cos²(x))³sin(x)cos⁶(x)dx

Apply u-substitution: ∫ −u⁶(1−u²)³du

=∫ −u⁶(1−u²)³du

Expand −u⁶(1−u²)³: −u⁶+3u⁸−3u¹⁰+u¹²

=∫ −u⁶+3u⁸−3u¹⁰+u¹²du

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

=−∫ u⁶du+∫ 3u⁸du−∫ 3u¹⁰du+∫ u¹²du

∫ u⁶du=u⁷7

∫ 3u⁸du=u⁹3

∫ 3u¹⁰du=3u¹¹11

∫ u¹²du=u¹³13

=−u⁷7 +u⁹3 −3u¹¹11 +u¹³13

Substitute back u=cos(x)

=−cos⁷(x)7 +cos⁹(x)3 −3cos¹¹(x)11 +cos¹³(x)13

Add a constant to the solution

=−cos⁷(x)7 +cos⁹(x)3 −3cos¹¹(x)11 +cos¹³(x)13 +C

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

integral of sin^7(x)cos^6(x)

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integral of sin^7(x)cos^6(x)

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