Reduction Formula Integrator

Math Calculus • Applications of Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

11. Reduction Formula Integrator

Computes recursive antiderivatives using reduction formulas (e.g., \(\int \sin^n(x)\,dx\), \(\int x^n e^x\,dx\)). Shows the recursion chain/tree and the final base-case result.

Inputs

Integral family

Choose a supported reduction-formula pattern.

Power \(n\)

Non-negative integer. (For demo presets, try \(n=4\).)

Options

Show steps Tree view Trig simplify Plot antiderivative

Plot \(x_{\min}\)

Plot \(x_{\max}\)

Presets

Click a preset to load & evaluate.

Ready

Graph

Drag to pan • wheel/pinch to zoom. Plots \(f(x)\) and (optionally) one antiderivative \(F(x)\) with \(C=0\).

x: 0, y: 0, zoom(px/unit): 60

Result

Choose a family and \(n\), then click Calculate.

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11. Reduction Formula Integrator — Theory

A reduction formula (recurrence) rewrites an integral with parameter \(n\) into an algebraic term plus another integral with a smaller parameter (like \(n-1\) or \(n-2\)). Repeating the step eventually reaches a base case that you already know.

\[ I_n=\int \sin^n(x)\,dx =-\frac{\cos x\,\sin^{n-1}x}{n}+\frac{n-1}{n}I_{n-2} \] \[ J_n=\int \cos^n(x)\,dx =\frac{\sin x\,\cos^{n-1}x}{n}+\frac{n-1}{n}J_{n-2} \]

How the \(\int \sin^n(x)\,dx\) reduction is derived

\[ I_n=\int \sin^{n-1}x\;\sin x\,dx \] Let \(u=\sin^{n-1}x\) and \(dv=\sin x\,dx\). Then \(du=(n-1)\sin^{n-2}x\cos x\,dx\) and \(v=-\cos x\). Integration by parts gives \[ I_n = -\sin^{n-1}x\cos x + (n-1)\int \sin^{n-2}x\cos^2x\,dx. \] Use \(\cos^2x = 1-\sin^2x\): \[ I_n = -\sin^{n-1}x\cos x + (n-1)\int \sin^{n-2}x\,dx - (n-1)\int \sin^nx\,dx. \] Recognize the last integral is \((n-1)I_n\), move it to the left: \[ nI_n = -\sin^{n-1}x\cos x + (n-1)I_{n-2} \quad\Rightarrow\quad I_n = -\frac{\cos x\,\sin^{n-1}x}{n} + \frac{n-1}{n}I_{n-2}. \]

Base cases

\[ I_0=\int 1\,dx=x,\qquad I_1=\int \sin x\,dx=-\cos x \] \[ J_0=\int 1\,dx=x,\qquad J_1=\int \cos x\,dx=\sin x \]

\(\int x^n e^x\,dx\) reduction

\[ K_n=\int x^n e^x\,dx \] Integration by parts with \(u=x^n\), \(dv=e^x dx\) gives \(du=nx^{n-1}dx\) and \(v=e^x\): \[ K_n = x^n e^x - n\int x^{n-1}e^x\,dx = x^n e^x - nK_{n-1}. \] Base case: \[ K_0=\int e^x\,dx=e^x. \]

Why the “trig simplify” step matches common closed forms

The reduction result for \(\sin^n\) or \(\cos^n\) often contains products like \(\sin^3x\cos x\) or \(\sin x\cos x\). These can be rewritten using identities (e.g., \(\sin x\cos x=\tfrac12\sin 2x\), \(\sin^2x=\tfrac{1-\cos 2x}{2}\)), producing multiple-angle sine/cosine terms and the compact textbook answers.

Frequently Asked Questions

What is a reduction formula in integration?

A reduction formula is a recurrence that rewrites an integral with parameter n in terms of algebraic terms plus another integral with a smaller parameter (such as n-1 or n-2). Repeating it eventually reaches a base case that is easy to integrate.

How does the reduction for integral sin^n(x) dx work?

It reduces the power by 2 each time, expressing I_n in terms of I_(n-2) plus a trigonometric product term. This continues until the base cases I_0 or I_1 are reached.

What are the base cases used by this reduction formula calculator?

For sine and cosine powers, typical base cases include integral 1 dx and integral sin(x) dx or integral cos(x) dx. For integral x^n e^x dx, the base case is integral e^x dx.

Why does the antiderivative plot use C = 0?

Any antiderivative differs only by a constant, so setting C = 0 provides one representative curve. The slope of the plotted antiderivative still matches the integrand.

What does the trig simplify option change in the result?

Trig simplify rewrites products like sin(x)cos(x) or higher powers using identities, often producing multiple-angle sine/cosine terms. This can make the final antiderivative match common textbook closed forms.

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Frequently Asked Questions

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