lambchamps
int. calculus
\[\int\limits_{}^{}\sin ^{2}x \cos ^{2}x \tan ^{2} x dx\]
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Freak7820
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Write them as
sin squared x = (1/2)(1 - cos 2x)
cos squared x = (1/2)(1 + cos 2x)
maybe..that'll be easier for you to integrate.
lambchamps
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can i do that, even if sin or cos is not raised two?
hartnn
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doesn't that simplify to sin^4 x when tan is written as sin/cos ?
SheldonEinstein
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Write tan^2 x = sin^2 x / cos^2 x
lambchamps
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yea
lambchamps
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i did that before
SheldonEinstein
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\[\large{\int { sin^4x dx}}\]
Can you do this?
lambchamps
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then i lost any of ideas after that
SheldonEinstein
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OK no problem, where are you lost?
hartnn
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can u integrate sin^2 x ?
lambchamps
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after i got the \[\int\limits_{?}^{?}\sin ^{4}xdx\]
SheldonEinstein
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Well if integration of sin x dx= -cosx + C
then that of sin^4 x dx = ?
lambchamps
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-cos^4x + C ??
hartnn
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write sin^4 x = (sin^2 x)(1-cos^2 x)
hartnn
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no, its not that direct
lambchamps
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then distribute?
hartnn
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= (sin^2 x)(1-cos^2 x) = sin^2 x - sin^2x cos^2 x
yes
hartnn
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= sin^2 x - sin^2x cos^2 x = sin^2 x - (1/4) (sin 2x)^2
lambchamps
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how did you get (1/4)(sin 2x)^2?
hartnn
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sin 2x = 2 sin x cos x
to get sin^2x cos^2 x , i squared the above .....
hartnn
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got that ? ^
and can u integrate sin^2 x - (1/4) (sin 2x)^2
lambchamps
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can you show me the whole process of that sin 2x = 2 sin x cos x
hartnn
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u want me to prove 2sin x cos x = sin 2x
or u wanted this :
sin 2x = 2 sin x cos x
(sin 2x)^2 = (2 sin x cos x)^2 = 4 sin^2x cos^2 x
so
sin^2 x cos^2 x = (1/4) (sin 2x)^2
hartnn
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so if u can integrate sin^2 x
you can also integrate sin^2 x - (1/4) (sin 2x)^2