Integrals
Class 12 Mathematics · Mathematics Part II · Chapter 1
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History
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Summary
Integration is the inverse process of differentiation. The indefinite integral \(\int f(x)\,dx=F(x)+C\) gives the family of antiderivatives of \(f\). This chapter builds a toolkit of methods: integration by substitution, integration using partial fractions, and integration by parts, \(\int u\,dv=uv-\int v\,du\). Standard forms for integrals of trigonometric, exponential and rational functions are tabulated. The definite integral \(\int_a^b f(x)\,dx\) is introduced as a limit of sums and evaluated using the Fundamental Theorem of Calculus, \(\int_a^b f(x)\,dx=F(b)-F(a)\), where \(F'=f\). Properties of definite integrals — additivity over intervals, symmetry over \([-a,a]\), and the king's rule \(\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx\) — simplify many evaluations.
Indefinite integrals and standard formsSubstitution and partial fractionsIntegration by partsDefinite integrals and the Fundamental TheoremProperties of definite integrals
Key terms
Indefinite integral
\(\int f(x)\,dx=F(x)+C\), the family of antiderivatives of \(f\).
Integration by substitution
Replacing \(x\) by a new variable to simplify the integrand.
Integration by parts
\(\int u\,dv=uv-\int v\,du\).
Partial fractions
Splitting a rational function into simpler fractions to integrate.
Definite integral
\(\int_a^b f(x)\,dx\), a number equal to the signed area under \(f\).
Fundamental Theorem of Calculus
\(\int_a^b f(x)\,dx=F(b)-F(a)\) where \(F'=f\).
Extra questions & answers
Try one from this chapter
The area under \(y=f(x)\) between \(x=a\) and \(x=b\) is: