Integrals

In this chapter: antiderivatives and the constant of integration, standard integrals, integration by substitution, integrals of special forms, partial fractions, integration by parts, definite integrals and the fundamental theorem of calculus, and properties of definite integrals.

Indefinite integrals

If d/dx F(x) = f(x), then F is an antiderivative (primitive) of f, and we write

Since the derivative of a constant is zero, x² + 1, x² − 5 and x² + 100 all have derivative 2x, so ∫2x dx is the whole family x² + C. Geometrically, these curves are vertical shifts of each other, with parallel tangents at any given x. Students often drop C in a hurry; in board answers that costs marks, and in MCQs the options sometimes differ only in how the constant has been absorbed.

Standard integrals (add + C to each)

Integration by substitution

If the integrand contains a function and (a multiple of) its derivative, put the function equal to t. For example, in ∫ 2x/(1 + x²) dx, put 1 + x² = t, so 2x dx = dt and the integral becomes ∫dt/t = log|t| + C = log(1 + x²) + C. A useful general form: ∫ f′(x)/f(x) dx = log|f(x)| + C. Trigonometric identities (sin²x = (1 − cos 2x)/2, products to sums) are often needed before a substitution becomes visible.

Integrals of special forms

For a quadratic ax² + bx + c in the denominator (or under a root), complete the square to reach one of these forms. For (px + q)/(ax² + bx + c), first write px + q = A·(derivative of the quadratic) + B.

Partial fractions

A proper rational function P(x)/Q(x) is split according to the factors of Q(x). If the degree of P is not less than that of Q, divide first.

Worked example: Evaluate ∫ dx/((x + 1)(x + 2)).
Solution: 1/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2). Putting x = −1 gives A = 1; putting x = −2 gives B = −1. So the integral is log|x + 1| − log|x + 2| + C = log |(x + 1)/(x + 2)| + C.

Integration by parts

Choose u as the function that becomes simpler on differentiating. The usual order of preference for u is ILATE: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. It is a guide, not a law.

Worked example: Evaluate ∫ x cos x dx.
Solution: Take u = x (algebraic) and v = cos x. Then ∫ x cos x dx = x sin x − ∫ 1 · sin x dx = x sin x + cos x + C. Check by differentiating: sin x + x cos x − sin x = x cos x.

Two results obtained by parts are used so often that they are worth remembering directly: ∫ e^x[f(x) + f′(x)] dx = e^x f(x) + C, and ∫ log x dx = x log x − x + C (take u = log x and v = 1).

Definite integrals

The fundamental theorem of calculus: if F is an antiderivative of a continuous function f on [a, b], then

When substituting in a definite integral, change the limits to the new variable as well. Then there is no need to return to x.

Worked example: Evaluate ∫₀¹ x e^x dx.
Solution: By parts, ∫ x e^x dx = x e^x − e^x. So the value is [x e^x − e^x]₀¹ = (e − e) − (0 − 1) = 1.

Properties of definite integrals

The splitting property handles |x − c| and [x]: break the interval where the expression inside changes sign or jumps.

Worked example: Evaluate I = ∫₀^π/2 sin x/(sin x + cos x) dx.
Solution: Replace x by π/2 − x: I = ∫₀^π/2 cos x/(cos x + sin x) dx. Adding the two forms, 2I = ∫₀^π/2 1 dx = π/2, so I = π/4. The same argument gives π/4 for sinⁿx/(sinⁿx + cosⁿx) and for √(sin x)/(√(sin x) + √(cos x)).

Common mistakes: (1) Leaving out + C in indefinite integrals. (2) Using xⁿ⁺¹/(n + 1) for n = −1; ∫dx/x is log|x|. (3) Changing the variable in a definite integral but keeping the old limits. (4) Choosing u badly in integration by parts, which makes the new integral harder than the old one. (5) Applying the odd-function property without checking: x² sin x is odd, but x sin x is even.

JEE and MHT‑CET focus

• Recognising the method quickly: substitution, special forms after completing the square, partial fractions or parts.
• The e^x[f(x) + f′(x)] pattern and integrals of the type ∫ e^ax sin bx dx.
• Partial fractions with repeated and quadratic factors.
• Definite integrals using the a + b − x property, even and odd functions, and periodicity.
• Integrals involving |x|, [x] and piecewise functions, by splitting the interval.

Practice questions