Reduction Formulae for Cosine Powers and Polynomial–Exponential Integrals

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What is a reduction formula?

A relation that expresses an integral with a higher parameter using lower-parameter integrals and possibly known terms.

What does a base case do in a reduction process?

It provides a directly evaluable integral where repeated reduction can stop.

Cosine-power reduction formula

For $I_n=\int \cos^n x\,dx$, $I_n=\frac{\sin x\cos^{n-1}x}{n}+\frac{n-1}{n}I_{n-2}$.

Polynomial–exponential reduction formula

For $I_n=\int x^n e^{ax}\,dx$ with $a\ne 0$, $I_n=\frac{x^n e^{ax}}{a}-\frac{n}{a}I_{n-1}$.

Back-substitution

Using lower-index results to build the value of the original integral.

Quiz questions

1. For $I_n=\int \cos^n x\,dx$, which recurrence reduces the exponent by two?Answer: $I_n=\frac{\sin x\cos^{n-1}x}{n}+\frac{n-1}{n}I_{n-2}$
2. For $I_n=\int x^n e^{ax}\,dx$, where $a\ne 0$, which recurrence follows from integration by parts?Answer: $I_n=\frac{x^n e^{ax}}{a}-\frac{n}{a}I_{n-1}$
3. For $I_n=\int x^n e^{ax}\,dx$, what should be done when $a=0$?Answer: Integrate the resulting power $x^n$ directly.
4. Repeatedly applying the cosine-power recurrence to $I_5$ reaches which base integral?Answer: $I_1=\int \cos x\,dx$
5. To derive a recurrence for $I_n=\int x^n e^{ax}\,dx$ by integration by parts, which choice is appropriate?Answer: $u=x^n$ and $dv=e^{ax}\,dx$