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integral of sin^4(x)cos^3(x)

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

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∫ sin⁴(x)cos³(x)dx

Solution

sin⁵(x)5 −sin⁷(x)7 +C

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Solution steps

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∫ sin⁴(x)cos³(x)dx

Rewrite using trig identities

=∫ (1−sin²(x))cos(x)sin⁴(x)dx

Apply u-substitution: ∫ u⁴(1−u²)du

=∫ u⁴(1−u²)du

Expand u⁴(1−u²): u⁴−u⁶

=∫ u⁴−u⁶du

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

=∫ u⁴du−∫ u⁶du

∫ u⁴du=u⁵5

∫ u⁶du=u⁷7

=u⁵5 −u⁷7

Substitute back u=sin(x)

=sin⁵(x)5 −sin⁷(x)7

Add a constant to the solution

=sin⁵(x)5 −sin⁷(x)7 +C

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

integral of sin^4(x)cos^3(x)

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