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amplitude of sin^2(x)

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

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

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∫ sin⁶(x)cos⁷(x)dx

Solution

sin⁷(x)7 −sin⁹(x)3 +3sin¹¹(x)11 −sin¹³(x)13 +C

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∫ sin⁶(x)cos⁷(x)dx

Rewrite using trig identities

=∫ (1−sin²(x))³cos(x)sin⁶(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=sin(x)

=sin⁷(x)7 −sin⁹(x)3 +3sin¹¹(x)11 −sin¹³(x)13

Add a constant to the solution

=sin⁷(x)7 −sin⁹(x)3 +3sin¹¹(x)11 −sin¹³(x)13 +C

Verify Result

=sin⁷(x)7 −sin⁹(x)3 +3sin¹¹(x)11 −sin¹³(x)13 +C

Order of operations three/four steps with parentheses Examples

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

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amplitude of sin^2(x)

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amplitude of sin^2(x)

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