Integration as the Reverse of Differentiation

Name: Integration as the Reverse of Differentiation
Author: StemPy Lab

You know that the derivative of $x^2$ is $2x$ . Now go the other way: which function has the derivative $2x$ ? Find antiderivatives, understand the constant of integration, learn the standard integrals and the rules, and then use substitution, with the help of trigonometric identities, to reach integrals that look hard. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Antiderivatives and the standard integrals
An antiderivative of $f$ is a function $F$ whose derivative is $f$ . If $F$ is one antiderivative, so is $F + C$ for any constant $C$ , and these are all of them. The whole family is written $\int f(x)\,dx = F(x) + C$ . Integration is differentiation run backwards, and every standard derivative gives a standard integral.
1 simulation1 practice question1 Python program3 problems2 Python labs
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Lesson 2about 25 min
Integration by substitution
Substitution undoes the chain rule. When the integrand contains an inner function $u = g(x)$ and, up to a constant, its derivative $g'(x)$ , the substitution turns $\int f(g(x))\,g'(x)\,dx$ into $\int f(u)\,du$ . Integrals of the form $\int f'(x)/f(x)\,dx$ give $\ln|f(x)|$ , and this gives the integrals of $\tan x$ , $\cot x$ , $\sec x$ and $\operatorname{cosec} x$ .
1 simulation1 practice question1 Python program4 problems
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Lesson 3about 25 min
Trigonometric identities that make a substitution possible
Many trigonometric integrals look unfamiliar until an identity rewrites them. Half-angle identities handle $\sin^2 x$ and $\cos^2 x$ . Peeling one factor off an odd power leaves a substitution. The product formulas turn $\sin mx\cos nx$ into a sum, and $\tan^2 x = \sec^2 x - 1$ turns powers of tangent into powers of secant.
1 simulation1 practice question1 Python program3 problems1 question set1 note
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Antiderivatives and Substitution: Practice Paper

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Integration as the Reverse of Differentiation

Course outline (3 lessons)

Antiderivatives and the standard integrals

Integration by substitution

Trigonometric identities that make a substitution possible

Test Your Learning

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