integral of (2xsin(x))/(cos^3(x))

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

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simplify solve for inverse tangent line

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∫ 2xsin(x)cos³(x) dx

Solution

xtan²(x)+x−tan(x)+C

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∫ 2xsin(x)cos³(x) dx

Take the constant out: ∫a·f(x)dx=a·∫f(x)dx

=2· ∫ xsin(x)cos³(x) dx

Rewrite using trig identities

=2· ∫ xtan(x)sec²(x)dx

Apply Integration By Parts: 12 xtan²(x)−∫ 12 tan²(x)dx

=2(12 xtan²(x)−∫ 12 tan²(x)dx)

∫ 12 tan²(x)dx=12 (−x+tan(x))

=2(12 xtan²(x)−12 (−x+tan(x)))

Simplify 2(12 xtan²(x)−12 (−x+tan(x))): xtan²(x)+x−tan(x)

=xtan²(x)+x−tan(x)

Add a constant to the solution

=xtan²(x)+x−tan(x)+C

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

integral of (2xsin(x))/(cos^3(x))

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