Definite Integral Notes for Class 12 & IIT JEE

A definite integral gives a number, the signed area between a curve and the x-axis, and most questions are solved by choosing the right property. These definite integral notes for Class 12 list the formulas you need, with short examples. In CBSE, the definite integral is part of the Integrals chapter. Writing an integral as the limit of a sum and Leibniz’s rule are mostly JEE-level, so check the current CBSE syllabus for your year.

What Is a Definite Integral? 

A definite integral computes the signed area between a function $f(x)$ and the x-axis over a closed interval $[a, b]$. Unlike an indefinite integral, it gives a specific number, not a function plus a constant. By the Fundamental Theorem of Calculus:

$$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$$

where $F(x)$ is any antiderivative of $f(x)$, and $a$ and $b$ are the lower and upper limits. Example:

$$\int_{0}^{2} (3x^2 + 1)\,dx = \left[x^3 + x\right]_0^2 = 10$$

Difference Between Definite and Indefinite Integrals

An indefinite integral gives a family of functions, $\int f(x)\,dx = F(x) + C$. A definite integral evaluates it between limits and returns a number. The constant $C$ cancels, so it is not written.

Definite Integral Notes for Class 12

The pages below are handwritten definite integral notes for Class 12. Use them along with the formulas above for quick revision.

Properties of Definite Integrals for Class 12 and JEE

• $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$ (the variable of integration does not matter).
• $\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$.
• $\displaystyle\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$.
• $\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$.
• $\displaystyle\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$ if $f$ is even, and $0$ if $f$ is odd.
• $\displaystyle\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx$ if $f(2a - x) = f(x)$, and $0$ if $f(2a - x) = -f(x)$.
• For a function with period $T$: $\displaystyle\int_0^{nT} f(x)\,dx = n\int_0^T f(x)\,dx$.

King’s Property of Definite Integrals

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

This is the most used property in JEE. Add the two forms of the integral, and the denominator often cancels. Example: let $I = \displaystyle\int_0^{\pi/2}\dfrac{\sin x}{\sin x + \cos x}\,dx$. Using the property, $I = \displaystyle\int_0^{\pi/2}\dfrac{\cos x}{\sin x + \cos x}\,dx$, so $2I = \displaystyle\int_0^{\pi/2} 1\,dx = \dfrac{\pi}{2}$ and $I = \dfrac{\pi}{4}$.

Even and Odd Functions

Check the symmetry first when the limits are $-a$ and $a$. For example, $\displaystyle\int_{-\pi/2}^{\pi/2}\sin^3 x\,dx = 0$ because $\sin^3 x$ is odd.

Evaluating a Definite Integral by Substitution

When you substitute, change the limits at the same time. Example: for $\displaystyle\int_0^1 2x\,e^{x^2}\,dx$, put $t = x^2$, so $dt = 2x\,dx$ and the limits become $0$ to $1$:

$$\int_0^1 e^t\,dt = e - 1$$

Wallis’s Formula

$$\int_0^{\pi/2}\sin^n x\,dx = \int_0^{\pi/2}\cos^n x\,dx = \dfrac{(n-1)(n-3)\cdots}{n(n-2)\cdots}\times\begin{cases}\dfrac{\pi}{2} & n \text{ even}\\[4pt] 1 & n \text{ odd}\end{cases}$$

Limit of a Sum as a Definite Integral

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

Example: $\displaystyle\lim_{n\to\infty}\dfrac{1}{n}\sum_{r=1}^{n}\dfrac{r}{n} = \int_0^1 x\,dx = \dfrac{1}{2}$.

Leibniz’s Rule (JEE Advanced)

$$\dfrac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)$$

Area Under a Curve Using a Definite Integral

The area between $y = f(x)$, the x-axis and the lines $x = a$ and $x = b$ is

$$A = \int_a^b |f(x)|\,dx$$

Split the integral where the curve crosses the x-axis, because area below the axis counts as negative in $\int f(x)\,dx$. For two curves with $f(x) \ge g(x)$ on $[a, b]$:

$$A = \int_a^b \left[f(x) - g(x)\right]dx$$

Example: the area under $y = x^2$ from $x = 0$ to $x = 3$ is $\displaystyle\int_0^3 x^2\,dx = 9$.

Common Mistakes in Definite Integral Notes for Class 12

• Forgetting to change the limits after substitution.
• Missing the sign change when the limits are swapped.
• Using the even or odd property when the limits are not symmetric.
• Treating a signed integral as area when the curve goes below the x-axis.

How to Revise Definite Integral Notes for Class 12

• Memorise the properties and the conditions where each applies.
• When the integrand has $\sin x$ and $\cos x$ in a fraction, try the King’s property first.
• Check symmetry and period before you integrate.
• For the rest of the syllabus, see our Class 12 Maths revision notes.