CAL-17

Power integral

∫ x^n dx = x^{n+1}/(n+1) + C, n ≠ −1.

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IntegralsPower integral

Governing equation

\int_0^x t^n\,dt=\dfrac{x^{n+1}}{n+1}

$x \in \left[0.05, 10\right]$

where

n: Exponent (—)
x: Upper limit (—)
I: Integral (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\int_0^x t^n\,dt=\dfrac{x^{n+1}}{n+1}

$x \in \left[0.05, 10\right]$

$x = 0.05$ $I = 4.17\times 10^{-5}$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.05,\quad x_{\max} = 10

$x \in \left[0.05, 10\right]$

$x = 0.05$ $x = 0.05$

Along the curve

③

Response curve — green

\int_0^x t^n\,dt=\dfrac{x^{n+1}}{n+1}

$x \in \left[0.05, 10\right]$

$x = 0.05$ $I = 4.17\times 10^{-5}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}x}

$x \in \left[0.05, 10\right]$

$x = 0.113$ $I^{\prime} = 0.014$

Along the curve

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-17 — Power integral) is the form associated with Power integral. Working symbols: $n$ , $x$ $\rightarrow$ $I$ . The inverse of the power rule. n = −1 is ln|x|.

Purpose

Purpose: compute

I

from

n

x

in Calculus via

\int_0^x t^n\,dt=\dfrac{x^{n+1}}{n+1}

∫ x^n dx = x^{n+1}/(n+1) + C, n ≠ −1. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 2.000\,\mathrm{—}

x = 3.000\,\mathrm{—}

, the governing relation

\int_0^x t^n\,dt=\dfrac{x^{n+1}}{n+1}

yields

I = 9.0000\,\mathrm{—}

. Evaluate the definite integral from 0 to x (n > −1). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Exponent n2 —Upper limit x3 —

Outputs

Integral I9.0000 —

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Narration of this film

Evaluate the definite integral from 0 to x (n > −1).

The inverse of the power rule. n = −1 is ln|x|.

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