integral of sec^4x

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

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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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∫ sec⁴(x)dx

Solution

tan(x)+tan³(x)3 +C

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∫ sec⁴(x)dx

Rewrite using trig identities

=∫ (1+tan²(x))sec²(x)dx

Apply u-substitution: ∫ 1+w²dw

=∫ 1+w²dw

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

=∫ 1dw+∫ w²dw

∫ 1dw=w

∫ w²dw=w³3

=w+w³3

Substitute back w=tan(x)

=tan(x)+tan³(x)3

Add a constant to the solution

=tan(x)+tan³(x)3 +C

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

integral of sec^4x

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