(adsbygoogle = window.adsbygoogle || []).push({}); The Same Integral... Two Different Answers!!!

1. The problem statement, all variables and given/known data

[tex] \int \frac{\tan x}{\cos^5 x} \,dx [/tex]

3. The attempt at a solution

Solution 1

[tex] \int \frac{\tan x}{\cos^5 x} \,dx = \int \frac{\sin x}{\cos^6 x} [/tex]

[tex]u = \cos x \,\,\,\, du = -\sin x \,dx [/tex]

[tex]= -\int u^-^5 \,du = \frac{1}{4 u^4} + C [/tex]

Answer 1:[tex] \frac{1}{4 \cos^4 x} + C [/tex]

Solution 2

[tex] \int \frac{\tan x}{\cos^5 x} \,dx = \int \tan x \sec^4 x \,dx = \int \tan x (1 + \tan^2 x) \sec^2 x \,dx [/tex]

[tex]u = \tan x \,\,\,\, du = \sec^2 x \,dx [/tex]

[tex] = \int \(u + u^3) \,du = \frac{1}{2} u^2 + \frac{1}{4} u^4 + C [/tex]

Answer 2:[tex] \frac{1}{2} \tan^2 x + \frac{1}{4} \tan^4^ x + C[/tex]

Wich one is right and why?

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# Homework Help: The Same Integral Two Different Answers

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