integral of tan^4 (x)

[math]

1. integral of tan^4 (x) dx


Here's my work:

= integral cos^4 (x) / sin^4 (X)
Let u= sinx du/dx = cosx dx = du/cosx

cos ^4 (x) / u^4 * du/cosx = cos3 (x) / u^4 du
cosx (1 - sin^2 (x))


I then tried approaching the problem a different way (using integration by parts)

u= tan^4 (x) v' = 1
u' = 4sec^3 (x) v=x

x tan^4 (x) - integral of 4x sec^3 (x)


...

 

[galactus]

Reduction formula for tangent:

\(\displaystyle \L\\\int{tan^{n}(u)}du=\frac{tan^{n-1}(u)}{n-1}-\int{tan^{n-2}(u)}du\)