Chapter 7: Integrals

Integration is the reverse process of differentiation. If $\dfrac{d}{dx}F(x) = f(x)$, then $F$ is an antiderivative of $f$, and we write $\int f(x)\,dx = F(x) + C$. The constant of integration $C$ captures the fact that infinitely many functions share the same derivative , they differ only by a constant. This single insight powers a vast machinery: from finding the area under a curve to computing work done by a variable force, from solving differential equations in physics to evaluating probabilities in continuous distributions.

Two integrals live in this chapter. The indefinite integral $\int f(x)\,dx$ produces a family of functions. The definite integral $\int_a^b f(x)\,dx$ produces a number , interpreted geometrically as signed area between the curve $y = f(x)$ and the $x$-axis on $[a, b]$. The two are stitched together by the Fundamental Theorem of Calculus: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$. This theorem is the deepest result in elementary calculus.

For board examinations the chapter is a calculation marathon. You must memorise standard integrals (logs, exponentials, trigonometric, inverse-trigonometric forms), recognise when to substitute, when to integrate by parts, and when to decompose into partial fractions. The LIATE rule, the $e^x[f(x)+f'(x)]$ pattern, and the trigonometric substitutions are tools that come back repeatedly.

For JEE the difficulty climbs. You will see integrals where the substitution is non-obvious, integrands involving $\sqrt{ax^2+bx+c}$, and definite integrals whose evaluation hinges on a symmetry property. Properties like $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$ unlock problems that look impossible by direct antidifferentiation.

The prerequisites are differentiation rules from Chapter 5 and a comfort with algebraic manipulation. You must know the derivative of every elementary function instinctively , those derivatives, read backwards, are the standard integrals.

A word on aesthetics: integration is an art. Unlike differentiation (which has a deterministic algorithm), integration requires recognition. Two integrals that look similar may need very different techniques. The only way to develop the instinct is sustained practice , there is no shortcut.

What's inside

1. Indefinite integrals and standard forms , antiderivatives, table of standards.
2. Integration by substitution , $u$-substitution and trigonometric substitution.
3. Integration by parts , LIATE, $\int e^x[f+f']\,dx$ trick.
4. Partial fractions , decomposing rational functions.
5. Definite integral as a limit of a sum , the Riemann definition.
6. Fundamental Theorem of Calculus , evaluation via antiderivatives.
7. Properties of definite integrals , symmetry, additivity, king's rule.
8. Special integrals , $\int\frac{dx}{x^2 \pm a^2}$, $\int\sqrt{a^2 - x^2}\,dx$ family.

Key results / Formula card

How to read this chapter

Memorise the standard table cold. Then practise recognition: which technique fits which integral? Build a personal mental flowchart , see $\ln x$ or $\tan^{-1}x$? Try by parts. See a rational function? Try partial fractions. See $\sqrt{a^2 - x^2}$? Try $x = a\sin\theta$. The chapter rewards pattern recognition above all.