Evaluate the integral. sin^(7)x cos^(5)x dx

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Evaluate the integral. sin^(7)x cos^(5)x dx .
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Комментарии
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Couldn't figure out what to split up if both were odd powers, but this cleared it up. Thanks!

dozydanny
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You deserve more spotlight. This was so helpful!

wes
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THANK U FOR THIS SUPER CLEAR N CONCISE EXPLANATION !!!!

mefrefgiweuhef
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You really save me, I was trying and trying and finally I got the way to do it

iliketrainsguy
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Quick Question, why wouldn’t you split sin in this case? Is it still possible to integrate just with more difficulty?

marker
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Thank you so much but can you please tell me the software you used as the board??

clintonfosu
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Solution:
At first the indefinite integral:
∫sin^7(x)*cos^5(x)*dx =
= =
—————————————————————
Substitution: u = sin(x) du = cos(x)*dx
—————————————————-
= ∫u^7*(1-u²)*(1-u²)*du = ∫u^7*(1-2u²+u^4)*du
= ∫(u^7-2u^9+u^11)*du = u^8/8-u^10/5+u^12/12+C
=

Sample by deriving:

=
=
= (sin^7(x)*[1-sin²(x)]²*cos(x)
= (sin^7(x)*[cos²(x)]²*cos(x)
= sin^7(x)*cos^5(x)

Now the definite integral:
π/2
=
0

1/8-1/5+1/12 = (15-24+10)/120 = 1/120

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