1. ## Intergration With Trigonometry

$\displaystyle \int \frac{\sec^5 \theta}{\tan^2\theta}d\theta$

I'm having troubles starting this. I've tried substituting for tangent,splitting up the secants, and a couple other methods but I get stuck after a couple steps. How should i start out?

2. Originally Posted by twoteenine
$\displaystyle \int \frac{\sec^5 \theta}{\tan^2\theta}d\theta$

I'm having troubles starting this. I've tried substituting for tangent,splitting up the secants, and a couple other methods but I get stuck after a couple steps. How should i start out?
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3. $\displaystyle \frac{ \sec^5(x) }{ \tan^2(x) }$

$\displaystyle = \frac{ \sec^4(x) }{ \tan^2(x) } \cdot \sec(x)$

$\displaystyle = \frac{ ( \tan^2(x) + 1)^2 }{ \tan^2(x) }\cdot \sec(x)$

$\displaystyle = \frac{ \tan^4(x) + 2\tan^2(x) + 1}{\tan^2(x) } \cdot \sec(x)$

$\displaystyle = [ \tan^2(x) + 2 + \frac{1}{\tan^2(x)} ] \cdot \sec(x)$

$\displaystyle = \tan^2(x)\sec(x) + 2\sec(x) + \frac{1}{ \tan(x) \cdot \tan(x) \cdot \cos(x) }$

$\displaystyle = \tan^2(x)\sec(x) + 2\sec(x) + \csc(x)\cot(x)$

its integral $\displaystyle = \int \tan^2(x)\sec(x) ~dx + 2\int \sec(x)~dx + \int \csc(x)\cot(x) ~dx$

$\displaystyle = \int \tan^2(x)\sec(x) ~dx + 2\int \sec(x)~dx - \csc(x) + C$

Consider $\displaystyle I = \int \tan^2(x)\sec(x) ~dx$

$\displaystyle I = \int [ \sec^3(x) - \sec(x) ]~dx$ $\displaystyle (1)$

but integration by parts gives

$\displaystyle I = \sec(x) \cdot \tan(x) - \int \sec^3 (x)~dx$

therefore ,

$\displaystyle \int \sec^3(x) ~dx = \sec(x) \tan(x) - I$ $\displaystyle (2)$

By substituting $\displaystyle (2)$ into $\displaystyle (1)$

we have

$\displaystyle I = \sec(x) \tan(x) - I - \int \sec(x)~dx ~~~ \implies$

$\displaystyle I = \frac{1}{2} \sec(x) \tan(x) - \frac{1}{2} \int \sec(x)~dx$

The integral

$\displaystyle \int \frac{ \sec^5(x) }{ \tan^2(x) } ~dx =$

$\displaystyle \frac{1}{2} \sec(x) \tan(x) + \frac{3}{2} \int \sec(x)~dx - \csc(x) + C$

$\displaystyle = \frac{1}{2} \sec(x) \tan(x) + \frac{3}{2} \cosh^{-1}[\sec(x)] - \csc(x) + C$