integral of tan^3(x)sec^5(x)

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(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

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∫ tan³(x)sec⁵(x)dx

Solution

sec⁷(x)7 −sec⁵(x)5 +C

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∫ tan³(x)sec⁵(x)dx

Rewrite using trig identities

=∫ (−1+sec²(x))tan(x)sec⁵(x)dx

Apply u-substitution: ∫ u⁴(u²−1)du

=∫ u⁴(u²−1)du

Expand u⁴(u²−1): u⁶−u⁴

=∫ u⁶−u⁴du

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

=∫ u⁶du−∫ u⁴du

∫ u⁶du=u⁷7

∫ u⁴du=u⁵5

=u⁷7 −u⁵5

Substitute back u=sec(x)

=sec⁷(x)7 −sec⁵(x)5

Add a constant to the solution

=sec⁷(x)7 −sec⁵(x)5 +C

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

integral of tan^3(x)sec^5(x)

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