integral of tan^4(x)sec^6(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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∫ tan⁴(x)sec⁶(x)dx

Solution

tan⁵(x)5 +2tan⁷(x)7 +tan⁹(x)9 +C

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∫ tan⁴(x)sec⁶(x)dx

Rewrite using trig identities

=∫ sec²(x)(1+tan²(x))²tan⁴(x)dx

Apply u-substitution: ∫ u⁴(1+u²)²du

=∫ u⁴(1+u²)²du

Expand u⁴(1+u²)²: u⁴+2u⁶+u⁸

=∫ u⁴+2u⁶+u⁸du

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

=∫ u⁴du+∫ 2u⁶du+∫ u⁸du

∫ u⁴du=u⁵5

∫ 2u⁶du=2u⁷7

∫ u⁸du=u⁹9

=u⁵5 +2u⁷7 +u⁹9

Substitute back u=tan(x)

=tan⁵(x)5 +2tan⁷(x)7 +tan⁹(x)9

Add a constant to the solution

=tan⁵(x)5 +2tan⁷(x)7 +tan⁹(x)9 +C

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

integral of tan^4(x)sec^6(x)

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