Class 12 Mathematics Chapter 7 Integrals — Formulas & Key Points

These formulas form the backbone of all integration work. Memorise them as automatic responses. Every formula assumes the integrand is a function of x and includes the arbitrary constant C for indefinite integrals. The power rule is the most frequently used formula, applicable whenever you see x raised to a rational or real exponent. Remember that the integral of 1/x is the natural logarithm, not a power function. The constant multiple rule allows you to factor out coefficients before integrating. When you see a polynomial, break it term by term and apply the power rule individually to each term. These formulas appear directly or after simplification in 70 percent of Board exam integration problems, so fluency here saves precious exam minutes and reduces silly errors during long calculations.

Trigonometric integrals appear in 2-3 questions every Board exam. Memorise the six basic forms perfectly, including signs. A common error is mixing up the signs for cos x and sec²x versus their negatives for csc²x and cot x. Notice the symmetry: integrals of sine and cosine are straightforward, but integrals of tangent and cotangent involve logarithms. The secant squared and cosecant squared integrals are direct antiderivatives of differentiation results you learned in Chapter 5 Continuity and Differentiability. When you encounter products or powers of trig functions, use identities like sin²x + cos²x = 1 or 1 + tan²x = sec²x to rewrite the integrand into standard forms. These formulas are also essential building blocks for integration by parts and substitution problems involving trigonometric expressions later in the chapter and in differential equations.

Exponential and logarithmic functions integrate cleanly because of their unique derivative properties. The integral of eˣ is simply eˣ plus the constant, making it one of the easiest functions to integrate. For the general exponential function aˣ, you must divide by the natural logarithm of the base a. This formula is less commonly tested but appears in competitive exams and occasionally in Board long-answer questions. The integral of 1/x is the natural logarithm, which you already saw in the algebraic section, but it warrants repetition because it bridges algebraic and transcendental functions. When you encounter expressions like e^(kx) where k is a constant, remember to apply the chain rule in reverse: the integral becomes (1/k)e^(kx). These formulas combine with substitution when the exponent or argument is a composite function, so practice identifying when to set u equal to the exponent or the argument.

Inverse trigonometric integrals arise from specific rational and irrational forms. These six formulas must be memorised exactly, including the denominators and the constants a. The first formula gives arcsin when you see a squared constant minus x squared in the square root. The second formula gives arctan when you see a squared constant plus x squared in the denominator without a square root. The third formula gives arcsec when you see x times the square root of x squared minus a squared in the denominator. Many students confuse these forms, so make flash cards or a mnemonic. The key is pattern recognition: identify the algebraic structure (square root of difference, sum of squares, product of x and square root) and match it to the correct inverse trig function. These forms often appear after substitution or partial fraction decomposition, especially in definite integral problems where limits are given. Practicing 10-15 problems of each type builds the muscle memory needed for Board exam speed and accuracy.

Substitution is the workhorse technique for CBSE Class 12 integration problems. The idea is to replace a composite function or a complicated expression with a single variable, say t, to simplify the integrand into a standard form. After integrating with respect to t, substitute back the original expression. The key skill is choosing the right substitution. Look for an inner function whose derivative is present elsewhere in the integrand. For definite integrals, transform the limits when you substitute so you do not need to substitute back. Common substitutions include setting t equal to the denominator, the argument of a trigonometric function, or the expression under a square root. Another powerful transformation is trigonometric substitution, where you replace x with a sin θ, a tan θ, or a sec θ to eliminate square roots involving quadratic expressions. Partial fractions decompose a rational function into simpler fractions when the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first, then apply partial fractions to the remainder term.

Integration by parts is the product rule of differentiation in reverse. Use it when the integrand is a product of two functions, one easy to differentiate (call it u) and the other easy to integrate (call it dv). The formula is ∫ u dv = u·v - ∫ v du. Choosing u and dv correctly is critical. The ILATE rule is a priority mnemonic: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Pick u in that order of preference and let the rest be dv. For example, in ∫ x·sin x dx, x is Algebraic and sin x is Trigonometric, so set u = x and dv = sin x dx. Then du = dx and v = -cos x, giving u·v - ∫ v du = x·(-cos x) - ∫ (-cos x) dx = -x cos x + sin x + C. Sometimes you need to apply integration by parts twice, especially with products like x²·eˣ or eˣ·sin x. In rare cases, applying integration by parts twice in a loop yields an equation you can solve algebraically for the original integral. This technique appears in 1-2 Board exam questions every year, often combined with substitution or standard forms.

Definite integrals have powerful properties that simplify evaluation, especially when the function or limits have symmetry. Property 1: ∫[a to b] f(x) dx = ∫[a to b] f(t) dt; the dummy variable does not matter. Property 2: ∫[a to b] f(x) dx = -∫[b to a] f(x) dx; reversing limits flips the sign. Property 3: ∫[a to b] f(x) dx = ∫[a to c] f(x) dx + ∫[c to b] f(x) dx; split the interval. Property 4: ∫[0 to a] f(x) dx = ∫[0 to a] f(a-x) dx; replace x with a minus x. This is powerful when f(x) plus f(a minus x) simplifies. Property 5: If f(x) is even (f(-x) = f(x)), then ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx. If f(x) is odd (f(-x) = -f(x)), then ∫[-a to a] f(x) dx = 0. Property 6: ∫[0 to 2a] f(x) dx = ∫[0 to a] f(x) dx + ∫[0 to a] f(2a-x) dx. These properties often convert a difficult integral into a trivial one, saving 5-10 minutes in the Board exam. Board setters love setting problems where recognizing symmetry or applying the f(a-x) substitution transforms the integrand into a constant or a simple algebraic form. Practice these properties with past Board papers to build intuition for when to apply which property.

The Fundamental Theorem of Calculus (FTC) is the bridge between differentiation and integration. It has two parts. Part 1: If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x). This says that the derivative of an integral with variable upper limit is the integrand evaluated at that limit. Part 2: If F is an antiderivative of f (meaning F'(x) = f(x)), then ∫[a to b] f(x) dx = F(b) - F(a). This is the evaluation formula you use every time you compute a definite integral. First find the indefinite integral (antiderivative), then substitute the upper limit, subtract the result of substituting the lower limit. The notation is F(x) evaluated from a to b, written as [F(x)]ₐᵇ or F(x)|ₐᵇ. The FTC justifies why we can ignore the constant of integration in definite integrals: F(b) + C minus (F(a) + C) cancels the C. Memorise the statement and practice applying Part 2 in every definite integral problem. Also remember that if f is continuous on [a, b], the integral from a to b exists and equals F(b) - F(a) for any antiderivative F.

Mastering terminology is essential for writing precise Board exam answers and understanding question wording. An indefinite integral or antiderivative is a family of functions F(x) + C whose derivative is the given function f(x). The arbitrary constant C accounts for the fact that derivatives of constants are zero. A definite integral is the limit of Riemann sums, representing the net signed area between the curve y = f(x) and the x-axis from x = a to x = b. The lower limit a and upper limit b are the bounds of integration. The integrand is the function f(x) being integrated. The variable of integration is x (or whichever variable appears in dx). An improper integral has one or both limits infinite, or the integrand has a discontinuity in the interval; these are introduced in higher courses but may appear as extension problems. A primitive or antiderivative is another name for the indefinite integral. The process of finding an integral is called integration or quadrature. Understanding these terms helps you parse problem statements and write answers that earn full method marks even if you make a small computational error.

ILATE mnemonic for integration by parts: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential — choose u in that priority order. King Property mnemonic: When you see limits 0 to a, try replacing x with a minus x and add the two forms; if lucky, terms cancel and you get a constant. Sign mistakes: integral of sin x is minus cos x (negative), integral of cos x is plus sin x (positive); integral of csc²x is minus cot x, integral of sec²x is plus tan x. Forgetting the constant: In indefinite integrals, always write plus C at the end; Board examiners deduct half a mark for this omission. Limits transformation in substitution: If you substitute u = g(x) in a definite integral, transform both limits; new lower limit is g(a), new upper limit is g(b); do not substitute back. Absolute value in logarithms: Always write ln|u| not ln(u) unless you know u is positive; this prevents domain errors. Partial fractions: Factor the denominator completely first; if the numerator degree is greater than or equal to the denominator degree, perform polynomial division before decomposing. Arctan and arcsin domain: Make sure the argument is within the domain of the inverse trig function; for arcsin, argument must be between minus one and one.

Example 1 (Standard Form): Evaluate ∫ (5x⁴ + 3/x - 2eˣ) dx. Split term by term: ∫ 5x⁴ dx + ∫ (3/x) dx - ∫ 2eˣ dx. Apply power rule, logarithm and exponential formulas: 5·(x⁵/5) + 3·ln|x| - 2·eˣ + C = x⁵ + 3ln|x| - 2eˣ + C. Example 2 (Substitution): Evaluate ∫ sin³x·cos x dx. Set u = sin x, then du = cos x dx. The integral becomes ∫ u³ du = (u⁴)/4 + C = (sin⁴x)/4 + C. Example 3 (Definite Integral with Property): Evaluate ∫[-π to π] x·sin x dx. Check: f(x) = x·sin x, f(-x) = (-x)·sin(-x) = (-x)·(-sin x) = x·sin x, so f is even. By property: ∫[-π to π] = 2∫[0 to π] x·sin x dx. Now integrate by parts: u = x, dv = sin x dx, du = dx, v = -cos x. ∫ x·sin x dx = x·(-cos x) - ∫ (-cos x) dx = -x·cos x + sin x. Evaluate from 0 to π: [-π·cos π + sin π] - [0·cos 0 + sin 0] = [-π·(-1) + 0] - [0 + 0] = π. So ∫[-π to π] = 2π. These examples cover standard forms, substitution and definite integrals with symmetry, three pillars of Board exam integration problems. Practice these methods until you can execute each in under three minutes.

Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n≠-1. Reciprocal: ∫ (1/x) dx = ln|x| + C. Exponential: ∫ eˣ dx = eˣ + C. Sine and Cosine: ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C. Sec²: ∫ sec²x dx = tan x + C. Csc²: ∫ csc²x dx = -cot x + C. Arcsin: ∫ dx/√(a²-x²) = sin⁻¹(x/a) + C. Arctan: ∫ dx/(a²+x²) = (1/a)tan⁻¹(x/a) + C. Integration by parts: ∫ u dv = uv - ∫ v du. ILATE for choosing u: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. FTC Part 2: ∫[a to b] f(x) dx = F(b) - F(a). King Property: ∫[0 to a] f(x) dx = ∫[0 to a] f(a-x) dx. Even function: ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx. Odd function: ∫[-a to a] f(x) dx = 0. Substitution: u=g(x), du=g'(x) dx; transform limits in definite integrals. Partial fractions: factor denominator, split into A/(x-r) terms, solve for coefficients. Always add constant C in indefinite integrals. Check signs for trig integrals. Write absolute value in logarithms. These formulas and tips cover 95 percent of Board exam integration questions; revise this box the night before your exam and again in the reading time.