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domäne x^3-4x

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∫ sin⁵(x)cos²(x)dx

Solution

−cos³(x)3 +2cos⁵(x)5 −cos⁷(x)7 +C

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∫ sin⁵(x)cos²(x)dx

Umschreiben mit Hilfe von Trigonometrie-Identitäten

=∫ (1−cos²(x))²sin(x)cos²(x)dx

Wende U-Substitution an: ∫ −u²(1−u²)²du

=∫ −u²(1−u²)²du

Multipliziere aus −u²(1−u²)²: −u²+2u⁴−u⁶

=∫ −u²+2u⁴−u⁶du

Wende die Summenregel an: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx

=−∫ u²du+∫ 2u⁴du−∫ u⁶du

∫ u²du=u³3

∫ 2u⁴du=2u⁵5

∫ u⁶du=u⁷7

=−u³3 +2u⁵5 −u⁷7

Setze in u=cos(x) ein

=−cos³(x)3 +2cos⁵(x)5 −cos⁷(x)7

Füge eine Konstante zur Lösung hinzu

=−cos³(x)3 +2cos⁵(x)5 −cos⁷(x)7 +C

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

domäne x^3-4x

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