anonymous
  • anonymous
∫ 7tan^5xdx
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
\[\int\limits7 \tan^5x dx\]
Psymon
  • Psymon
Turn it into \[\int\limits_{}^{}7(\tan ^{2}x)(\tan ^{2}x)(tanx)dx\]Then take the two tan^2x terms and turn them into secants using the identity: \[\tan ^{2}x + 1 = \sec ^{2}x \]After that you should be able to multiply things out and integrate.
anonymous
  • anonymous
Ok, I got that far but my final answer is wrong

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anonymous
  • anonymous
my final answer was 7/4 sec^4x- tan^2x + ln Isec xI
Psymon
  • Psymon
Yeah, looks like we might have to try it only converting one of the tan^2's \[7\int\limits_{}^{}(\sec ^{2}x-1)(\sec ^{2}x-1)(tanx)\] \[7\int\limits_{}^{}(\sec ^{4}x-2\sec ^{2}x-1)(tanx)dx\] \[7[\int\limits_{}^{}\sec ^{4}xtanxdx-2\int\limits_{}^{}\sec ^{2}xtanxdx-\int\limits_{}^{}tanxdx]\] The first two integrals let u = secx u = secx, du = secxtanxdx, dx = du/(secxtanx) \[7[\int\limits_{}^{}u^{4}tanx*\frac{ du }{ utanx }-2\int\limits_{}^{}u ^{2}tanx*\frac{ du }{ utanx }-\int\limits_{}^{}tanxdx\]
Psymon
  • Psymon
the tanx's cancel and one u cancels in each one. So youd have u^3 for the firstone, u^1 for the second one. The derivative of tanx = -ln|secx|
anonymous
  • anonymous
is what you wrote the final answer?
Psymon
  • Psymon
*integral of tanx my bad. And I wrote out everything but the actual integration of the u's and the tanx.
anonymous
  • anonymous
that looks confusing. Is there a way to simplify this more?
Psymon
  • Psymon
\[7[\int\limits_{}^{}u ^{3}-2\int\limits_{}^{}u-\int\limits_{}^{}tanx]\] Once I cancel things out I have that. Without actually integrating, that's as simple as I can get it down to.
anonymous
  • anonymous
ok
anonymous
  • anonymous
I still don't see how my answer was wrong then....
Psymon
  • Psymon
Hmm... \[7(\frac{ u ^{4} }{ 4 }-u ^{2}-\ln|secx|)\] \[\frac{ 7\sec ^{4}x }{ 4 }-7\sec ^{2}x-7\ln|secx|+C\]
anonymous
  • anonymous
ok, i didnt have the 7's in front of all those
Psymon
  • Psymon
And you had a different sign in front of ln|secx|
anonymous
  • anonymous
it says that is right. thank you so much!
anonymous
  • anonymous
oh yea! i did
Psymon
  • Psymon
Awesome, glad ya got it ^_^

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