integral of sin^5(x)cos^6(x)

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

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log_□

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ddx

≥	≤	·	÷	`x`^◦	(☐)	\|☐\|	(`f` ◦ `g`)	`f`(`x`)	ln	`e`^☐
(☐)^′	∂∂`x`	∫_□^□	lim	∑	sin	cos	tan	cot	csc	sec

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∫ sin⁵(x)cos⁶(x)dx

Solution

−cos⁷(x)7 +2cos⁹(x)9 −cos¹¹(x)11 +C

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∫ sin⁵(x)cos⁶(x)dx

Rewrite using trig identities

=∫ (1−cos²(x))²sin(x)cos⁶(x)dx

Apply u-substitution: ∫ −u⁶(1−u²)²du

=∫ −u⁶(1−u²)²du

Expand −u⁶(1−u²)²: −u⁶+2u⁸−u¹⁰

=∫ −u⁶+2u⁸−u¹⁰du

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

=−∫ u⁶du+∫ 2u⁸du−∫ u¹⁰du

∫ u⁶du=u⁷7

∫ 2u⁸du=2u⁹9

∫ u¹⁰du=u¹¹11

=−u⁷7 +2u⁹9 −u¹¹11

Substitute back u=cos(x)

=−cos⁷(x)7 +2cos⁹(x)9 −cos¹¹(x)11

Add a constant to the solution

=−cos⁷(x)7 +2cos⁹(x)9 −cos¹¹(x)11 +C

Verify Result

=−cos⁷(x)7 +2cos⁹(x)9 −cos¹¹(x)11 +C

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

integral of sin^5(x)cos^6(x)

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integral of sin^5(x)cos^6(x)

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