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

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

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∫ sin³(x)cos⁴(x)dx

Solution

−cos⁵(x)5 +cos⁷(x)7 +C

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∫ sin³(x)cos⁴(x)dx

Rewrite using trig identities

=∫ (1−cos²(x))sin(x)cos⁴(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=cos(x)

=−cos⁵(x)5 +cos⁷(x)7

Add a constant to the solution

=−cos⁵(x)5 +cos⁷(x)7 +C

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

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

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