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What is the integral of #int tan^5(x)*sec^4(x)dx#?

1 Answer

Jul 5, 2016

#= 1/8tan^8(x) + 1/6tan^6(x) + C#

Explanation:

#int dx qquad tan^5(x)*sec^4(x)#

#= int dx qquad tan^5(x)*color{red}{sec^2(x)}*sec^2(x)#

using well known identity....
#= int dx qquad tan^5(x)*color{red}{(tan^2(x) + 1)}*sec^2(x)#

#= int dx qquad (tan^7(x) + tan^5(x))*sec^2(x)#

#= int dx qquad tan^7(x)*sec^2(x) + tan^5(x)*sec^2(x)#

#= 1/8tan^8(x) + 1/6tan^6(x) + C#

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