What's the integral of #int (tanx)^5*(secx)^4dx#?

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What's the integral of #int (tanx)^5*(secx)^4dx#?

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Answer 1 · 2018-03-16T14:56:17

#I=tan^6x/6+tan^8x/8+c#

Explanation:

We know that,
#color(red)(int[f(x)]^n*f^'(x)dx=[f(x)]^(n+1)/(n+1)+c)#
#f(x)=tanx=>f^'(x)=sec^2x#
So,
#I=int(tanx)^5*(secx)^4dx=int(tanx)^5(sec^2x)(secx)^2dx#
#I=int(tanx)^5(1+tan^2x)(sec^2x)dx#
#I=int(tanx)^5sec^2xdx+int(tanx)^7sec^2xdx#
#I=int(tanx)^5d/(dx)(tanx)dx+int(tanx)^7d/(dx)(tanx)dx#
#I=(tanx)^(5+1)/(5+1)+(tanx)^(7+1)/(7+1)+c#
#I=tan^6x/6+tan^8x/8+c#

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What's the integral of #int (tanx)^5*(secx)^4dx#?

1 Answer

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