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integral of (cos^2(x))/(sin^3(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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∫ cos²(x)sin³(x) dx

Solution

18 (tan²(x2 )−4ln|tan(x2 )|−cot²(x2 ))+C

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∫ cos²(x)sin³(x) dx

Apply u-substitution: ∫ (−u²+1)²4u³ du

=∫ (−u²+1)²4u³ du

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

=14 · ∫ (−u²+1)²u³ du

Expand (−u²+1)²u³ : u−2u +1u³

=14 · ∫ u−2u +1u³ du

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

=14 (∫ udu−∫ 2u du+∫ 1u³ du)

∫ udu=u²2

∫ 2u du=2ln|u|

∫ 1u³ du=−12u²

=14 (u²2 −2ln|u|−12u² )

Substitute back u=tan(x2 )

=14 (tan²(x2 )2 −2ln|tan(x2 )|−12tan²(x2 ) )

Simplify 14 (tan²(x2 )2 −2ln|tan(x2 )|−12tan²(x2 ) ): 18 (tan²(x2 )−4ln|tan(x2 )|−cot²(x2 ))

=18 (tan²(x2 )−4ln|tan(x2 )|−cot²(x2 ))

Add a constant to the solution

=18 (tan²(x2 )−4ln|tan(x2 )|−cot²(x2 ))+C

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

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

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