What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

What is an indefinite integral?

\(\int f(x)\,dx=F(x)+C\), the set of all antiderivatives of \(f\), where \(C\) is an arbitrary constant.

State the integration by parts formula.

\(\int u\,dv=uv-\int v\,du\); choose \(u\) by the ILATE preference order.

State the Fundamental Theorem of Calculus.

If \(F'=f\) on \([a,b]\), then \(\int_a^b f(x)\,dx=F(b)-F(a)\).

Evaluate \(\int x^n\,dx\) for \(n\neq-1\).

\(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\).

Evaluate \(\int \dfrac{1}{x}\,dx\).

\(\int \dfrac{1}{x}\,dx=\log|x|+C\).

State the property of \(\int_a^b f(x)\,dx\) under reversal of limits.

\(\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\).

What is \(\int_{-a}^{a} f(x)\,dx\) for an odd function?

It is \(0\); for an even function it equals \(2\int_0^a f(x)\,dx\).

When are partial fractions used?

To integrate a proper rational function by decomposing it into a sum of simpler fractions whose integrals are known.

Evaluate \(\int e^x\,dx\) and \(\int \sin x\,dx\).

\(\int e^x\,dx=e^x+C\) and \(\int \sin x\,dx=-\cos x+C\).

Integrals

Q: State the king's property of definite integrals.

\(\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx\).

Class 12 Mathematics · Mathematics Part II · Chapter 1

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CBSE Class 12 · Mathematics

Integrals: 35 marks across the four papers

Marks booster + cheatsheet: key concepts in 2 lines each
23 marks in the four papers: Vector Algebra
Integral tables with properties P0–P7
3 methods for differential equations
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CBSE Class 12 · Mathematics

Every Part I question from 2025 + the SQP

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35 marks: Calculus, shared with Part II
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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:

Flashcards & interactive practice