Can reduction formula be applied on $\int \cos^n x \: dx$ when n is a negative integer?

Question

The reduction formula for integration of $\cos^n x$ is:

$$ \int \cos^n x \: dx=\frac1n \cos^{n-1} x\sin x+\frac{n-1}n \int \cos^{n-2} x \: dx $$

But if $n$ is a negative integer like $-1, -2,-3,\ldots $ then can this reduction formula still be applied?

Answer 1

You may read the relation $$ \int \cos^n x \: dx=\frac1n \cos^{n-1} x\sin x+\frac{n-1}n \int \cos^{n-2} x \: dx \tag1 $$ in the form $$ \int \cos^{n-2} x \: dx=-\frac1{n-1} \cos^{n-1} x\sin x+\frac n{n-1} \int \cos^{n} x \: dx. \tag2 $$ Then, for negative integers like $n=-1, -2,-3,\cdots, $ by a recurrence, you will end up with considering $$ \int \cos^{-2} x \: dx \quad \text{or} \quad \int \cos^{-1} x \: dx. \tag3 $$

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Two examples.

We just use $(2)$.

$$ \begin{align} \int \cos^{-4} x \: dx=\int \cos^{-2\color{red}{-2}} x \: dx&=\frac13 \cos^{-3} x\sin x+\frac 23 \int \cos^{-2} x \: dx\\\\ &=\frac13 \cos^{-3} x\sin x+\frac 23 \tan x +C \end{align} $$

and

$$ \begin{align} \int \cos^{-3} x \: dx=\int \cos^{-1\color{red}{-2}} x \: dx&=\frac12 \cos^{-2} x\sin x+\frac 12 \int \cos^{-1} x \: dx\\\\ &=\frac12 \cos^{-2} x\sin x+\frac 12 \log \left|\tan (x/2+\pi/4)\right| +C \end{align} $$