CalculusVery High Yield (3-4 Qs in JEE)

Definite Integration & Properties Formulas

Definite integrals, symmetry properties, King's rule, periodic functions, and differentiation under the integral sign.

8 Formulas•838+ Questions in Bank•

Concept Diagram

Definite Integral & Tangent Line

Geometric interpretation of ∫ f(x) dx as area under curve and derivative f'(x₀) as tangent slope.

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

Transforming x into (a + b - x) leaves the integral value invariant.

Examiner Tip: Add the new integral to the original integral I to cancel awkward denominators (2I technique).

\int_{-a}^a f(x) \, dx = \begin{cases} 2 \int_0^a f(x) \, dx & \text{if } f(-x) = f(x) \text{ (even)} \\ 0 & \text{if } f(-x) = -f(x) \text{ (odd)} \end{cases}

Symmetry test across origin drastically reduces definite integral calculations.

Examiner Tip: Check odd symmetry immediately for symmetric limits [-a, a].

\int_0^{nT} f(x) \, dx = n \int_0^T f(x) \, dx, \quad \int_a^{a + T} f(x) \, dx = \int_0^T f(x) \, dx

If f(x + T) = f(x), integrals over multiple periods pull factor n outside.

Examiner Tip: The integral over any full period T is completely independent of the starting limit a.

\frac{d}{dx} \int_{\alpha(x)}^{\beta(x)} f(t) \, dt = f(\beta(x)) \beta'(x) - f(\alpha(x)) \alpha'(x)

Differentiating a definite integral with variable boundary limits with respect to x.

Examiner Tip: Crucial for applying L'Hôpital's Rule on definite integral indeterminate forms.

\lim_{n \to \infty} \sum_{r=1}^n \frac{1}{n} f\left(\frac{r}{n}\right) = \int_0^1 f(x) \, dx

Converts infinite Riemann series summations into standard definite integrals (r/n -> x, 1/n -> dx).

Examiner Tip: Lower bound = lim (r_min/n), Upper bound = lim (r_max/n).

\int_0^{\pi/2} \sin^n x \, dx = \frac{(n-1)(n-3)\dots}{(n)(n-2)\dots} \times K \quad \left(K = \frac{\pi}{2} \text{ if } n \text{ even, } 1 \text{ if odd}\right)

Instant evaluation of definite powers of sine and cosine over [0, π/2].

Examiner Tip: Saves up to 5 minutes compared to standard reduction integration.

\int_0^{2a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(2a - x) = f(x) \\ 0 & \text{if } f(2a - x) = -f(x) \end{cases}

Splits interval [0, 2a] based on reflection symmetry across the midpoint x = a.

Examiner Tip: Essential for integrating ∫₀^π x f(sin x) dx: first remove x using King's property, then apply half-integral rule.

m(b - a) \le \int_a^b f(x)\,dx \le M(b - a), \quad \left(\int_a^b f(x)g(x)\,dx\right)^2 \le \left(\int_a^b [f(x)]^2\,dx\right)\left(\int_a^b [g(x)]^2\,dx\right)

Bounding bounds for definite integrals where m and M are infimum and supremum of f(x) on [a, b].

Examiner Tip: Modulus inequality: |∫ f(x) dx| ≤ ∫ |f(x)| dx.

Frequently tested misconceptions in Definite Integration & Properties

•Missing the chain-rule derivative terms when applying the Leibniz Rule with variable upper and lower limits.
•Applying King's property without adding the resulting integral back to the original (2I = ...).