foci y=4x^2-x-3

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∫ cos⁹(x)dx

Solution

sin(x)−4sin³(x)3 +6sin⁵(x)5 −4sin⁷(x)7 +sin⁹(x)9 +C

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Solution steps

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∫ cos⁹(x)dx

Rewrite using trig identities

=∫ (1−sin²(x))⁴cos(x)dx

Apply u-substitution: ∫ (1−u²)⁴du

=∫ (1−u²)⁴du

Expand (1−u²)⁴: 1−4u²+6u⁴−4u⁶+u⁸

=∫ 1−4u²+6u⁴−4u⁶+u⁸du

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

=∫ 1du−∫ 4u²du+∫ 6u⁴du−∫ 4u⁶du+∫ u⁸du

∫ 1du=u

∫ 4u²du=4u³3

∫ 6u⁴du=6u⁵5

∫ 4u⁶du=4u⁷7

∫ u⁸du=u⁹9

=u−4u³3 +6u⁵5 −4u⁷7 +u⁹9

Substitute back u=sin(x)

=sin(x)−4sin³(x)3 +6sin⁵(x)5 −4sin⁷(x)7 +sin⁹(x)9

Add a constant to the solution

=sin(x)−4sin³(x)3 +6sin⁵(x)5 −4sin⁷(x)7 +sin⁹(x)9 +C

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

foci y=4x^2-x-3

Solution

Solution steps

Number Line

Graph

1	−4
4	−7

3	1	1
1	3	−1
2	4	1