Chapter 7 — Integrals

What is Integration?

Integration is a fundamental operation in calculus that can be thought of in two closely related ways:

• Inverse of differentiation: If F'(x) = f(x), then F(x) is called an antiderivative or an indefinite integral of f(x). We write F(x) = ∫ f(x) dx + C (C is an arbitrary constant).
• Limit of sums / area interpretation: A definite integral ∫_a^b f(x) dx gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Formally it is the limit of Riemann sums: ∫_a^b f(x) dx = lim_n→∞ Σ f(x_i*) Δx.

Indefinite vs Definite Integrals

• Indefinite integral: ∫ f(x) dx = F(x) + C, represents a family of antiderivatives.
• Definite integral: ∫_a^b f(x) dx = F(b) − F(a) (when F is an antiderivative of f). This gives a number (signed area).

Fundamental Theorem of Calculus (FTC)

• Part 1 (Connection): If F(x) = ∫_a^x f(t) dt and f is continuous, then F'(x) = f(x). So integration followed by differentiation returns the original function.
• Part 2 (Evaluation): If F is any antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a). This is the main tool to compute definite integrals.

When is a function integrable?

A bounded function that is continuous almost everywhere (or piecewise continuous) on [a,b] is Riemann integrable. For CBSE Class 12, typical functions considered are continuous or piecewise continuous.

Basic idea of computation

To compute indefinite integrals you use standard formulas (power rule, basic trigonometric integrals, exponential, logarithmic) and techniques (substitution, integration by parts). For definite integrals, you find an antiderivative and apply the FTC; alternatively, evaluate limits of sums for area problems.

What is an indefinite integral? An indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). It is written as ∫f(x) dx = F(x) + C, where C is an arbitrary constant (the constant of integration). The indefinite integral represents the family of all antiderivatives of f.

Basic ideas and properties

• Linearity: ∫[a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx.
• Power rule (inverse of differentiation): for n ≠ −1, ∫ x^n dx = x^{n+1}/(n+1) + C.
• Special case: ∫ x^{-1} dx = ∫ (1/x) dx = ln|x| + C.
• Substitution idea: If u = g(x) and du = g'(x) dx, then ∫ f(g(x)) g'(x) dx = ∫ f(u) du (useful for chain-rule reversal).
• Interpretation: The indefinite integral gives a family of curves that differ by a vertical translation. In applied contexts it recovers accumulated quantity from a rate (e.g., position from velocity).

How to use the standard formulas: Recognize parts of the integrand that match a known formula, factor constants, split sums, and use substitution when you see a composition with a derivative factor. For products where substitution is not direct, use integration by parts: ∫ u dv = uv − ∫ v du.

Integration is the inverse process of differentiation and is used to find areas, accumulated quantities and antiderivatives. 'Methods of Integration' refers to a set of techniques used to evaluate indefinite integrals (antiderivatives) and definite integrals when a direct antiderivative is not obvious.

Main methods (with when to use):

• Substitution (u-substitution) — use when integrand is of the form f(g(x))·g'(x) so a change of variable simplifies the integral.
• Integration by parts — useful for products of functions (polynomial·exponential, polynomial·log, polynomial·trig). Based on the product rule for derivatives.
• Partial fraction decomposition — for rational functions (quotient of polynomials); decompose into simpler rational terms whose integrals are known.
• Trigonometric integrals and identities — for integrands involving powers or products of sin x, cos x, tan x, etc.; use identities and substitutions like sin^2 x = (1−cos2x)/2.
• Trigonometric substitution — for integrands with root expressions like sqrt(a^2 − x^2), sqrt(a^2 + x^2), sqrt(x^2 − a^2); substitute x = a sin θ, x = a tan θ, x = a sec θ respectively.
• Weierstrass substitution (t = tan(x/2)) — converts rational functions of sin x and cos x into rational functions of t, useful for complicated trig integrals.
• Reduction formulas — recursive relations for integrals of powers (e.g., ∫sin^n x dx) allowing reduction to lower powers.

Strategy: try to simplify the integrand (algebra/trig identities), look for a derivative inside the integrand (for substitution), choose parts wisely for integration by parts (let u be the function that simplifies when differentiated), and break rational functions into partial fractions when degree of numerator < denominator.

Definition (Riemann sum): If f is a function defined on [a,b], its definite integral is the limit of Riemann sums (when the limit exists):

∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i*) Δx, where Δx = (b−a)/n and x_i* ∈ [x_i−1, x_i].

Geometric meaning: The definite integral gives the signed area under the curve y = f(x) between x = a and x = b. Areas above the x-axis contribute positively, below contribute negatively.

Fundamental Theorem of Calculus (two parts):

• Part I (Relationship of integral and derivative): If f is continuous on [a,b] and F(x) = ∫_a^x f(t) dt, then F is differentiable and F′(x) = f(x).
• Part II (Evaluation): If F is any antiderivative of f on [a,b] (i.e. F′(x)=f(x)), then ∫_a^b f(x) dx = F(b) − F(a).

Basic properties (used often in computations):

• Linearity: ∫_a^b [αf(x)+βg(x)] dx = α∫_a^b f(x) dx + β∫_a^b g(x) dx.
• Additivity: ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx for a≤c≤b.
• Reversal: ∫_b^a f(x) dx = −∫_a^b f(x) dx.
• Zero width: ∫_a^a f(x) dx = 0.
• Monotonicity: If f(x) ≥ g(x) on [a,b], then ∫_a^b f(x) dx ≥ ∫_a^b g(x) dx.
• |∫ f| ≤ ∫ |f| (absolute value inequality).
• Even/Odd functions (symmetric limits): If f is even, ∫_−a^a f(x) dx = 2∫₀^a f(x) dx. If f is odd, ∫_−a^a f(x) dx = 0.

Common evaluation techniques: Use an antiderivative (FTC II). Useful methods to find antiderivatives include substitution (change of variable) and integration by parts (definite form). For difficult integrals, numerical methods (Riemann sums, trapezoidal rule, Simpson's rule) approximate the value.

Mean Value Theorem for Integrals: If f is continuous on [a,b], there exists c in (a,b) such that ∫_a^b f(x) dx = f(c)(b−a). This gives the average value f_avg = (1/(b−a))∫_a^b f(x) dx.

Area between two curves: If f(x) ≥ g(x) on [a,b], area between them is ∫_a^b [f(x) − g(x)] dx.

Notes for CBSE Class 12: practice evaluating definite integrals by finding antiderivatives, using properties to simplify limits or symmetry, and interpreting integrals in application problems (area, displacement, probability).

Definite integrals have algebraic properties that make evaluation easier, especially when the integrand or interval has symmetry. Key ideas: linearity, additivity of intervals, change of limits sign, and symmetry about the origin or about the mid-point of the interval. Using symmetry often reduces an integral over a large interval to an integral over half the interval or sometimes shows the integral is zero.

Important symmetry types:

• Even functions: f(−x) = f(x). For any a > 0, ∫_−a^a f(x) dx = 2 ∫₀^a f(x) dx. Geometrically the area on the left and right of the y-axis are equal.
• Odd functions: f(−x) = −f(x). For any a, ∫_−a^a f(x) dx = 0. The positive area cancels the negative area because of antisymmetry.
• Mid-point or interval symmetry: On [a,b], many integrals simplify using the substitution x → a + b − x. In particular ∫_a^b f(x) dx = ∫_a^b f(a+b−x) dx, so sometimes f(x)+f(a+b−x) is simpler (or constant).

Why this matters: symmetry reduces computation and gives insight (for example, center-of-mass or expected value of a symmetric distribution often becomes zero). Combining properties with substitution or algebraic manipulation can turn a difficult integral into an easy one.

Practical tips:

• Check if f is even or odd before integrating on [−a,a].
• For integrals on [a,b] try the substitution x → a+b−x to detect mid-point symmetry.
• If f(x)+f(a+b−x) simplifies to a constant or simple function, evaluate the average times interval length.

Overview. A definite integral ∫_a^b f(x) dx gives the signed area under the curve y = f(x) from x = a to x = b. Key techniques transform a given integral into an easier one or exploit properties of the integrand or limits.

Main techniques.

• Fundamental Theorem of Calculus (FTC). If F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). Use FTC after finding an antiderivative.
• Substitution (Change of Variable). If x = g(t) is a smooth bijection on the interval, transform ∫_a^b f(x) dx to ∫_{g^{-1}(a)}^{g^{-1}(b)} f(g(t)) g'(t) dt. This simplifies integrands that are compositions.
• Integration by Parts for Definite Integrals. Use ∫_a^b u dv = [u v]_a^b − ∫_a^b v du to reduce products of functions, e.g., x e^x, x sin x.
• Symmetry and the Property f(x) ↔ f(a+b−x). ∫_a^b f(x) dx = ∫_a^b f(a+b−x) dx. For a = −b and even/odd functions: if f is even, ∫_{−a}^{a} f(x) dx = 2 ∫_0^a f(x) dx; if f is odd, ∫_{−a}^{a} f(x) dx = 0.
• Decomposition and Partial Fractions. Break rational functions into simpler fractions whose antiderivatives are known.
• Reduction Formulas. Use recurrence relations to reduce powers or parameters step by step.
• Use of Definite Integral Properties. Linearity, additivity of intervals, reversal of limits, and comparison inequalities help simplify or estimate integrals.
• Known Standard Integrals. Recognize forms like ∫ 1/(1+x^2) dx = arctan x, ∫ e^x dx = e^x, ∫ sin x dx = −cos x, etc.
• Numerical Methods (if not integrable in closed form). Trapezoidal rule and Simpson's rule approximate definite integrals when antiderivatives are hard or impossible to find.

Strategy to choose a technique. Look for substitution patterns (inner derivative present), product forms (use parts), symmetry of interval or integrand (use even/odd or f(a+b−x)), rational functions (try partial fractions), and standard integral shapes.

Important remarks. For improper integrals (infinite limits or singularities), treat as limits: ∫_a^{∞} f(x) dx = lim_{R→∞} ∫_a^R f(x) dx, and check convergence.

Introduction
The definite integral gives the net signed area under a curve. Using limits of Riemann sums, area of a plane region bounded by curves can be computed by evaluating definite integrals. In Class 12 you use integrals to find:

• Area under a single curve and the x-axis between x = a and x = b.
• Area between two curves y = f(x) and y = g(x) over [a, b].
• Area computed by integrating with respect to y when convenient.

Derivation idea (brief)
Split the interval [a,b] into n small subintervals of width Δx. Approximate the area by summing areas of rectangles of height f(x_i*) and base Δx. Taking the limit as n → ∞ gives the definite integral A = ∫_a^b f(x) dx (if f(x) ≥ 0 on [a,b]).

Sign consideration
A definite integral ∫_a^b f(x) dx gives signed area: parts above the x-axis contribute positive area, parts below contribute negative. For geometric area always take absolute values or split intervals where sign changes.

Area between two curves
If f(x) ≥ g(x) on [a,b], the area of the region between them is
A = ∫_a^b [f(x) − g(x)] dx. If f and g cross within [a,b], find intersection points, split the integral at those x-values, and add absolute values of each piece.

Integrating with respect to y
When curves are better described as x = φ(y), x = ψ(y) on y ∈ [c,d], the area between them is
A = ∫_c^d [x_right(y) − x_left(y)] dy. This is useful when vertical strips would require splitting but horizontal strips give a single integral.

Procedure / Steps to find area between curves

• Sketch the curves and find intersection points (solve f(x)=g(x)).
• Decide which function is on top (greater) on each subinterval.
• Set up the integral(s): A = ∑ ∫ [top − bottom] dx (or dy).
• Evaluate the integral(s) and add results (use absolute values if needed).

Common mistakes to avoid

• Not checking which function is greater on the interval.
• Using signed integral value directly when asked for geometric area (use absolute values or split intervals where sign changes).
• For regions bounded by vertical and non-vertical lines, choose dx or dy appropriately to avoid unnecessary splitting.

What are Special Forms? In integration, certain integrands appear in standard patterns whose antiderivatives are immediate once you recognise the pattern. These are called special forms. They typically arise when the integrand contains a function and its derivative (or a constant multiple of it), or matches known elementary integral formulas (logarithmic, inverse trigonometric, exponential, trigonometric).

How to use them: Look for an inner function u = f(x) whose derivative f'(x) (or a constant multiple) is present in the integrand. Use substitution u = f(x) to convert the integral into a basic form. Always add the constant of integration C for indefinite integrals.

Common patterns (idea):

• If integrand is f'(x)/f(x) → result ln|f(x)| + C.
• If integrand is f'(x) (f(x))^n → result (f(x))^{n+1}/(n+1) + C for n ≠ -1.
• If integrand is f'(x) e^{f(x)} → result e^{f(x)} + C.
• If integrand is f'(x) [sin f(x)] or [cos f(x)] → use substitution to get cos or -sin forms.
• Standard trigonometric and inverse-trigonometric forms and rational forms (partial fractions) should be memorised.

Remarks:

• Constant multiple: If you have k f'(x)/f(x), factor out k: k ln|f(x)| + C.
• Not every integrand has an elementary antiderivative (e.g. e^{-x^2} has no elementary primitive; its definite integral is handled via Gaussian methods or special functions).