Integral of $\frac1{\cos^n x}$

Question

Hi guys I have already proven for an assignment that:

$$\int\cos(x)^n dx=\frac{1}{n}\cos(x)^{n−1}\sin(x) + \frac{n-1}{n}\int\cos(x)^{n−2}dx$$

Now we have been asked to calculate $$\int_0^{\pi/4}\frac{1}{\cos^4(x)}dx$$ theoretically make $n=-4$, however the expression below will only work for $n\geq2$ right?

Halp please

Answer 1

The identity holds unless integer $n\ne0$

First set $n=-2$

$$\int\cos(x)^{-2} dx=\frac{1}{-2}\cos(x)^{-2−1}\sin(x) + \frac{-2-1}{-2}\int\cos(x)^{-2−2}dx$$

$$\iff\frac32\int\cos(x)^{-4}dx=\int\cos(x)^{-2} dx+\frac12\cos(x)^{-3}\sin(x)$$

Now, $$\int\cos^{-2}x\ dx=\int\sec^2x\ dx=\tan x+K$$