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### Integral of Trig. Function, Power

```
Date: 7/19/96 at 11:25:22
Subject: Integral of Trig. Function, Power

Can you solve this one?
Integral of dx/cos^4 x

Thanks
Sg
```

```
Date: 7/19/96 at 17:12:26
From: Doctor Anthony
Subject: Re: Integral of Trig. Function, Power

I = INT[dx/cos^4(x)] = INT[sec^4(x)*dx]

= INT[(1+tan^2(x))*sec^2(x) dx]

Integrate by parts

I = (1+tan^2(x))*tan(x) - INT[tan(x)*2tan(x)*sec^2(x)*dx]

=  (1+tan^2(x))*tan(x) - 2*INT[tan^2(x)*sec^2(x)*dx]

=  (1+tan^2(x))*tan(x) - 2*Int[(sec^2(x)-1)*sec^2(x)*dx]

= tan(x)+tan^3(x) - 2*INT[{sec^4(x) - sec^2(x)}*dx]

I = tan(x)+tan^3(x) - 2*I + 2*INT[sec^2(x)*dx]

3I = tan(x)+tan^3(x) + 2*tan(x) + Const

I = (1/3)(tan(x)+tan^3(x)) + (2/3)tan(x) + C

=  tan(x) + (1/3)tan^3(x) + C

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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