Integral Calculus formulas for JEE Main & Advanced

23 formula entries for Integral Calculus, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Indefinite Integral

\int f (x) d x = F (x) + C

When to use. When finding the general antiderivative of a function without specified limits.

Trap. Forgetting the constant of integration C in indefinite integrals.

Definite Integral

\int_{a}^{b} f (x) d x = F (b) - F (a)

When to use. When computing area, net change, or evaluating integrals with limits.

Trap. Forgetting to subtract F(a) from F(b) (order matters).

Power Rule for Integration

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C (n \neq - 1)

\int \frac{1}{x} d x = \ln ∣ x ∣ + C

When to use. When integrating polynomial terms or any x^n form.

Trap. Applying power rule for n = -1 (gives ln|x|, not x^0/0).

Integration of Trigonometric Functions

\int \sin x d x = - \cos x + C

\int \cos x d x = \sin x + C

\int \sec^{2} x d x = \tan x + C

\int \csc^{2} x d x = - \cot x + C

\int \sec x \tan x d x = \sec x + C

\int \csc x \cot x d x = - \csc x + C

When to use. When integrating trigonometric functions directly or after substitution.

Trap. Forgetting the negative sign in ∫sin x dx and ∫csc^2 x dx.

Integration of Exponential and Logarithmic Functions

\int e^{x} d x = e^{x} + C

\int a^{x} d x = \frac{a^{x}}{\ln a} + C

\int \ln x d x = x \ln x - x + C

When to use. When integrating exponential or logarithmic functions directly.

Trap. Forgetting the division by ln a for a^x.

Integration by Substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

where

u = g (x)

When to use. When the integrand contains a function and its derivative (or a constant multiple).

Trap. Forgetting to change dx to du correctly.

Integration by Parts

\int u d v = u v - \int v d u

When to use. When the integrand is a product of two different types of functions (e.g., x e^x, x sin x, ln x).

Trap. Choosing u and dv incorrectly (e.g., picking dv that is hard to integrate).

Integration by Partial Fractions

For distinct linear factors:

\frac{A}{x - a} + \frac{B}{x - b}

For repeated linear:

\frac{A}{x - a} + \frac{B}{(x - a)^{2}}

For quadratic:

\frac{A x + B}{x^{2} + p x + q}

When to use. When integrating rational functions where denominator can be factored.

Trap. Not checking if degree of numerator ≥ denominator (perform polynomial division first).

Standard Integrals Involving Quadratic Expressions

\int \frac{d x}{x^{2} + a^{2}} = \frac{1}{a} \arctan \frac{x}{a} + C

\int \frac{d x}{\sqrt{a^{2} - x^{2}}} = \arcsin \frac{x}{a} + C

\int \frac{d x}{\sqrt{x^{2} \pm a^{2}}} = \ln ∣ x + \sqrt{x^{2} \pm a^{2}} ∣ + C

When to use. When the integrand involves $\sqrt{a^{2} - x^{2}}$ , $\sqrt{x^{2} \pm a^{2}}$ , or $\frac{1}{x^{2} + a^{2}}$ .

Trap. Forgetting the factor 1/a in arctan form.

Integration Using Trigonometric Identities

\sin^{2} x = \frac{1 - \cos 2 x}{2}

\cos^{2} x = \frac{1 + \cos 2 x}{2}

\sin x \cos x = \frac{1}{2} \sin 2 x

When to use. When integrating powers of sin and cos, or products like sin mx cos nx.

Trap. Not applying identities to reduce powers.

Integration of Irrational Functions

For

\sqrt{a^{2} - x^{2}}

x = a \sin θ

For

\sqrt{x^{2} + a^{2}}

x = a \tan θ

For

\sqrt{x^{2} - a^{2}}

x = a \sec θ

When to use. When the integrand contains $\sqrt{a^{2} - x^{2}}$ , $\sqrt{x^{2} \pm a^{2}}$ , or $\sqrt{a x + b}$ .

Trap. Forgetting to convert dx in terms of dθ.

Definite Integral Properties

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

Even:

\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x

Odd:

\int_{- a}^{a} f (x) d x = 0

When to use. When evaluating definite integrals, especially with symmetric limits or periodic functions.

Trap. Forgetting the sign change when swapping limits.

Leibniz Rule for Differentiation Under Integral Sign

\frac{d}{d t} \int_{a (t)}^{b (t)} f (x, t) d x = f (b (t), t) b^{'} (t) - f (a (t), t) a^{'} (t) + \int_{a (t)}^{b (t)} \frac{\partial f}{\partial t} d x

When to use. When evaluating integrals that depend on a parameter, especially in JEE advanced problems.

Trap. Forgetting to differentiate the limits.

Area Under Curves

Area =

\int_{a}^{b} ∣ f (x) - g (x) ∣ d x

When to use. When computing area between curves or under a curve.

Trap. Not taking absolute value when curves cross.

Integration as Limit of a Sum

\int_{a}^{b} f (x) d x = \lim_{n \to \infty} \frac{b - a}{n} \sum_{r = 1}^{n} f (a + r \frac{b - a}{n})

When to use. When evaluating limits of sums that resemble Riemann sums.

Trap. Not matching the sum form correctly to the integral.

Gamma Function and Beta Function

Γ (n) = \int_{0}^{\infty} x^{n - 1} e^{- x} d x

B (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x = \frac{Γ (m) Γ (n)}{Γ (m + n)}

When to use. When evaluating integrals involving powers and exponentials, or products of powers.

Trap. Forgetting that Γ(n+1)=n! only for positive integers.

Reduction Formulas

\int \sin^{n} x d x = - \frac{1}{n} \sin^{n - 1} x \cos x + \frac{n - 1}{n} \int \sin^{n - 2} x d x

\int \cos^{n} x d x = \frac{1}{n} \cos^{n - 1} x \sin x + \frac{n - 1}{n} \int \cos^{n - 2} x d x

When to use. When integrating high powers of trigonometric functions.

Trap. Forgetting the coefficient (n-1)/n.

Integration of Piecewise Functions

\int_{a}^{b} f (x) d x = \int_{a}^{c} f_{1} (x) d x + \int_{c}^{b} f_{2} (x) d x

where

f (x) = {\begin{cases} f_{1} (x) & x < c \\ f_{2} (x) & x \geq c \end{cases}

When to use. When the integrand is defined piecewise, e.g., |x|, floor functions, or mod functions.

Trap. Not identifying the breakpoints correctly.

Integration Using Euler's Substitution

a > 0

\sqrt{a x^{2} + b x + c} = t - \sqrt{a} x

c > 0

\sqrt{a x^{2} + b x + c} = x t + \sqrt{c}

When to use. When integrating rational functions with √(ax^2+bx+c) that cannot be handled by standard trigonometric substitutions.

Trap. Choosing the wrong Euler substitution based on sign of a or c.

Integration of Rational Functions of sin x and cos x

t = \tan \frac{x}{2}

\sin x = \frac{2 t}{1 + t^{2}}

\cos x = \frac{1 - t^{2}}{1 + t^{2}}

d x = \frac{2}{1 + t^{2}} d t

When to use. When integrating rational functions of sin x and cos x, e.g., ∫ dx/(a sin x + b cos x + c).

Trap. Forgetting the factor 2 in dx.

Integration of Irrational Functions of the Form ∫dx/√(ax^2+bx+c)

\int \frac{d x}{\sqrt{a x^{2} + b x + c}} = \frac{1}{\sqrt{a}} \ln ∣ 2 a x + b + 2 \sqrt{a (a x^{2} + b x + c)} ∣ + C

(for a>0)

When to use. When integrand is 1/√(quadratic) or similar.

Trap. Forgetting to complete the square correctly.

Integration of √(ax^2+bx+c)

\int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \arcsin \frac{x}{a} + C

\int \sqrt{x^{2} \pm a^{2}} d x = \frac{x}{2} \sqrt{x^{2} \pm a^{2}} \pm \frac{a^{2}}{2} \ln ∣ x + \sqrt{x^{2} \pm a^{2}} ∣ + C

When to use. When integrating √(quadratic) expressions.

Trap. Forgetting the ± sign in the logarithmic term.

Integration Using Special Substitutions (e.g., x = a cos^2 θ + b sin^2 θ)

x = a \cos^{2} θ + b \sin^{2} θ

Then

d x = 2 (b - a) \sin θ \cos θ d θ

When to use. When integrand contains √((x-a)(b-x)) or similar forms.

Trap. Not recognizing when this substitution is applicable.

Values to remember

$e \approx 2.718$ Integration of Exponential and Logarithmic Functions
$Γ (1 / 2) = \sqrt{π}$ Gamma Function and Beta Function

Read the full chapter: Integral Calculus for JEE, with what JEE asks and the common traps.