Indefinite Integral Notes for Class 12 & IIT JEE
eSaral Academic and Editorial Team
Summary
Indefinite Integral Notes for Class 12 cover the antiderivative and the constant of integration, the standard integrals, the special integrals of the form $\int \frac{dx}{x^2 \pm a^2}$ and $\int \sqrt{a^2 - x^2}\,dx$, and the methods of substitution, integration by parts, partial fractions and trigonometric identities. This page gives all the formulas in LaTeX, solved examples and handwritten notes for CBSE boards and JEE Main.
Integrals is Chapter 7 of NCERT Class 12 Mathematics, and Application of Integrals is Chapter 8. Indefinite integration is the base of the whole Calculus unit, which carries the largest share of marks in the CBSE paper. You also need it for the definite integrals, areas and differential equations that follow. These Indefinite Integral Notes for Class 12 follow the NCERT chapter, and they add the techniques that JEE problems need.
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
India's Best Exam Preparation for Class 12th - Download Now
Indefinite Integral Notes for Class 12: Chapter Overview and Syllabus
The CBSE chapter covers:
- Integration as the inverse process of differentiation
- Integration of a variety of functions by substitution, by partial fractions and by parts
- Evaluation of the standard integrals of the forms $\int \dfrac{dx}{x^2 \pm a^2}$, $\int \dfrac{dx}{\sqrt{x^2 \pm a^2}}$, $\int \dfrac{dx}{ax^2 + bx + c}$, $\int \dfrac{px + q}{ax^2 + bx + c}\,dx$, and the integrals $\int \sqrt{a^2 \pm x^2}\,dx$ and $\int \sqrt{x^2 - a^2}\,dx$
- The fundamental theorem of calculus and definite integrals (the next topic)
JEE Main lists integral as an antiderivative, the fundamental integrals of algebraic, trigonometric, exponential and logarithmic functions, and integration by substitution, by parts and by partial fractions. JEE Advanced uses the same techniques in harder combinations. Check the latest syllabi of CBSE and NTA (for JEE Main) and the IITs (for JEE Advanced).
Tip from the eSaral Maths team: Always check an indefinite integral by differentiating your answer. If the derivative gives back the original function, the answer is right, and it takes only a few seconds.
Antiderivative and Properties in Indefinite Integral Notes for Class 12
If $\dfrac{d}{dx}F(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$, and we write
$$\int f(x)\,dx = F(x) + C$$
where $C$ is the constant of integration. An indefinite integral is a family of functions, one for each value of $C$.
$$\int \big[a f(x) + b g(x)\big]\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$ $$\frac{d}{dx}\int f(x)\,dx = f(x), \qquad \int f'(x)\,dx = f(x) + C$$
Standard Integrals in Indefinite Integral Notes for Class 12
Function | Integral |
|---|---|
$x^n$ ($n \neq -1$) | $\dfrac{x^{n+1}}{n+1} + C$ |
$\dfrac{1}{x}$ | $\ln\lvert x \rvert + C$ |
$e^x$ | $e^x + C$ |
$a^x$ | $\dfrac{a^x}{\ln a} + C$ |
$\sin x$ | $-\cos x + C$ |
$\cos x$ | $\sin x + C$ |
$\sec^2 x$ | $\tan x + C$ |
$\operatorname{cosec}^2 x$ | $-\cot x + C$ |
$\sec x\tan x$ | $\sec x + C$ |
$\operatorname{cosec} x\cot x$ | $-\operatorname{cosec} x + C$ |
$\tan x$ | $\ln\lvert \sec x \rvert + C$ |
$\cot x$ | $\ln\lvert \sin x \rvert + C$ |
$\sec x$ | $\ln\lvert \sec x + \tan x \rvert + C$ |
$\operatorname{cosec} x$ | $\ln\lvert \operatorname{cosec} x - \cot x \rvert + C$ |
$\dfrac{1}{\sqrt{1 - x^2}}$ | $\sin^{-1}x + C$ |
$\dfrac{1}{1 + x^2}$ | $\tan^{-1}x + C$ |
$\dfrac{1}{x\sqrt{x^2 - 1}}$ | $\sec^{-1}x + C$ |
Special Integrals in Indefinite Integral Notes for Class 12
Integral | Result |
|---|---|
$\displaystyle\int \frac{dx}{x^2 + a^2}$ | $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + C$ |
$\displaystyle\int \frac{dx}{x^2 - a^2}$ | $\dfrac{1}{2a}\ln\left\lvert \dfrac{x - a}{x + a} \right\rvert + C$ |
$\displaystyle\int \frac{dx}{a^2 - x^2}$ | $\dfrac{1}{2a}\ln\left\lvert \dfrac{a + x}{a - x} \right\rvert + C$ |
$\displaystyle\int \frac{dx}{\sqrt{a^2 - x^2}}$ | $\sin^{-1}\dfrac{x}{a} + C$ |
$\displaystyle\int \frac{dx}{\sqrt{x^2 + a^2}}$ | $\ln\left\lvert x + \sqrt{x^2 + a^2} \right\rvert + C$ |
$\displaystyle\int \frac{dx}{\sqrt{x^2 - a^2}}$ | $\ln\left\lvert x + \sqrt{x^2 - a^2} \right\rvert + C$ |
$\displaystyle\int \sqrt{a^2 - x^2}\,dx$ | $\dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a} + C$ |
$\displaystyle\int \sqrt{x^2 + a^2}\,dx$ | $\dfrac{x}{2}\sqrt{x^2 + a^2} + \dfrac{a^2}{2}\ln\left\lvert x + \sqrt{x^2 + a^2} \right\rvert + C$ |
$\displaystyle\int \sqrt{x^2 - a^2}\,dx$ | $\dfrac{x}{2}\sqrt{x^2 - a^2} - \dfrac{a^2}{2}\ln\left\lvert x + \sqrt{x^2 - a^2} \right\rvert + C$ |
$\displaystyle\int e^{ax}\sin bx\,dx$ | $\dfrac{e^{ax}(a\sin bx - b\cos bx)}{a^2 + b^2} + C$ |
$\displaystyle\int e^{ax}\cos bx\,dx$ | $\dfrac{e^{ax}(a\cos bx + b\sin bx)}{a^2 + b^2} + C$ |
To integrate $\dfrac{1}{ax^2 + bx + c}$ or $\dfrac{1}{\sqrt{ax^2 + bx + c}}$, complete the square, and then use one of the formulas above.
Integration by Substitution in Indefinite Integral Notes for Class 12
If $u = g(x)$, then $du = g'(x)\,dx$, and
$$\int f\big(g(x)\big)\,g'(x)\,dx = \int f(u)\,du$$
Two useful forms are:
$$\int \frac{f'(x)}{f(x)}\,dx = \ln\lvert f(x) \rvert + C, \qquad \int \big[f(x)\big]^n f'(x)\,dx = \frac{\big[f(x)\big]^{n+1}}{n+1} + C$$
When to use it: look for a function and its derivative in the integrand, such as $x\,e^{x^2}$ or $\dfrac{\cos x}{\sin x}$. After the substitution, write the answer in terms of $x$ again.
Integration by Parts in Indefinite Integral Notes for Class 12
$$\int u\,v\,dx = u\int v\,dx - \int\left(\frac{du}{dx}\int v\,dx\right)dx$$
Choosing $u$ (the ILATE rule): pick $u$ in the order Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential, and take the other function as $v$.
A special form:
$$\int e^x\big[f(x) + f'(x)\big]\,dx = e^x f(x) + C$$
Repeated integration by parts: for $\int x^2 e^x\,dx$, apply the rule twice. For integrals like $\int e^{ax}\sin bx\,dx$, apply it twice, and the original integral appears again on the right side, so you can solve for it.
Integration by Partial Fractions in Indefinite Integral Notes for Class 12
Use partial fractions for a rational function $\dfrac{P(x)}{Q(x)}$. If the degree of $P$ is not less than the degree of $Q$, divide first.
Factor in the denominator | Partial fraction |
|---|---|
$(x - a)$ | $\dfrac{A}{x - a}$ |
$(x - a)^2$ | $\dfrac{A}{x - a} + \dfrac{B}{(x - a)^2}$ |
$(x - a)(x - b)$ | $\dfrac{A}{x - a} + \dfrac{B}{x - b}$ |
$(x^2 + bx + c)$ (irreducible) | $\dfrac{Ax + B}{x^2 + bx + c}$ |
Trigonometric Integrals for JEE in Indefinite Integral Notes for Class 12
Use these identities to reduce powers of $\sin x$ and $\cos x$:
$$\sin^2 x = \frac{1 - \cos 2x}{2}, \qquad \cos^2 x = \frac{1 + \cos 2x}{2}, \qquad \sin x\cos x = \frac{\sin 2x}{2}$$
- For $\int \sin^m x\cos^n x\,dx$, if $m$ is odd, substitute $u = \cos x$. If $n$ is odd, substitute $u = \sin x$.
- For integrals like $\int \dfrac{dx}{a + b\cos x}$ or $\int \dfrac{dx}{a\sin x + b\cos x}$, the substitution $t = \tan\dfrac{x}{2}$ turns the integrand into a rational function.
Reduction formula (JEE Advanced):
$$I_n = \int \sin^n x\,dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\,I_{n-2}$$
Key Formulas for Indefinite Integration
Concept | Formula |
|---|---|
Power rule | $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C,\ n \neq -1$ |
Logarithmic form | $\displaystyle\int \dfrac{f'(x)}{f(x)}\,dx = \ln\lvert f(x) \rvert + C$ |
By parts | $\displaystyle\int u\,v\,dx = u\int v\,dx - \int\left(\dfrac{du}{dx}\int v\,dx\right)dx$ |
Special form | $\displaystyle\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C$ |
$\tan^{-1}$ form | $\displaystyle\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + C$ |
$\sin^{-1}$ form | $\displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac{x}{a} + C$ |
Square root form | $\displaystyle\int \sqrt{a^2 - x^2}\,dx = \dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a} + C$ |
Solved Examples from Indefinite Integral Notes for Class 12
Example 1 (standard integrals): Find $\displaystyle\int (3x^2 + 2x + 1)\,dx$.
$\displaystyle\int (3x^2 + 2x + 1)\,dx = x^3 + x^2 + x + C$.
Example 2 (trigonometric identity): Find $\displaystyle\int \sin^2 x\,dx$.
$\sin^2 x = \dfrac{1 - \cos 2x}{2}$, so $\displaystyle\int \sin^2 x\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$.
Example 3 (substitution): Find $\displaystyle\int x\,e^{x^2}\,dx$.
Put $u = x^2$, so $du = 2x\,dx$. Then $\displaystyle\int x\,e^{x^2}\,dx = \dfrac{1}{2}\int e^u\,du = \dfrac{e^{x^2}}{2} + C$.
Example 4 (by parts): Find $\displaystyle\int x\,e^x\,dx$ and $\displaystyle\int \ln x\,dx$.
For the first, take $u = x$ and $v = e^x$: $\displaystyle\int x\,e^x\,dx = x\,e^x - e^x + C = e^x(x - 1) + C$. For the second, take $u = \ln x$ and $v = 1$: $\displaystyle\int \ln x\,dx = x\ln x - x + C$.
Example 5 (partial fractions): Find $\displaystyle\int \frac{dx}{(x - 1)(x - 2)}$.
$\dfrac{1}{(x-1)(x-2)} = \dfrac{1}{x - 2} - \dfrac{1}{x - 1}$, so the integral is $\ln\lvert x - 2 \rvert - \ln\lvert x - 1 \rvert + C = \ln\left\lvert \dfrac{x - 2}{x - 1} \right\rvert + C$.
Example 6 (special form): Find $\displaystyle\int e^x(\sin x + \cos x)\,dx$.
Take $f(x) = \sin x$, so $f'(x) = \cos x$. Then $\displaystyle\int e^x(\sin x + \cos x)\,dx = e^x\sin x + C$.
Example 7 (completing the square): Find $\displaystyle\int \frac{dx}{x^2 + 4x + 13}$.
$x^2 + 4x + 13 = (x + 2)^2 + 3^2$, so the integral is $\dfrac{1}{3}\tan^{-1}\dfrac{x + 2}{3} + C$.
Example 8 (logarithmic form): Find $\displaystyle\int \frac{2x + 3}{x^2 + 3x + 5}\,dx$.
The numerator is the derivative of the denominator, so the integral is $\ln\lvert x^2 + 3x + 5 \rvert + C$.
Common Mistakes in Indefinite Integration Problems
Mistake | Correct approach |
|---|---|
Forgetting the constant $C$ | Add $+C$ to every indefinite integral |
Writing $\displaystyle\int \tan x\,dx = \ln\lvert \cos x \rvert + C$ (forgetting the minus sign) | $\displaystyle\int \tan x\,dx = -\ln\lvert \cos x \rvert + C = \ln\lvert \sec x \rvert + C$ |
Using $\displaystyle\int \dfrac{1}{x}\,dx = \dfrac{x^0}{0}$ | The power rule fails for $n = -1$. The answer is $\ln\lvert x \rvert + C$. |
Choosing $u$ against ILATE and getting a harder integral | Follow the order Inverse, Logarithmic, Algebraic, Trigonometric, Exponential |
Forgetting to change $dx$ after a substitution | Replace $dx$ using $du = g'(x)\,dx$, and return to $x$ at the end |
Not dividing first in an improper rational function | Divide the numerator by the denominator before using partial fractions |
Not checking the answer | Differentiate the result, and compare it with the integrand |
How to Study Indefinite Integral Notes for Class 12
- Learn the table of standard integrals, and write it from memory every day for a week.
- Practise the methods in this order: standard forms, substitution, trigonometric identities, partial fractions, by parts.
- Solve the exercises of Chapter 7 with eSaral’s NCERT Solutions for Class 12 Maths Chapter 7. You can download the textbook from eSaral’s NCERT Class 12 Maths book page.
- Revise all the Class 12 chapters with eSaral’s Class 12 Maths notes, and solve eSaral’s JEE Main chapter-wise PYQs on integration.
Frequently Asked Questions
Is integration by parts important for Class 12 boards?
Yes. Integration by parts is a standard technique in the CBSE Class 12 chapter on Integrals, and it is tested as a short or a long answer question. Learn the ILATE rule, and also the special form $\displaystyle\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C$, which gives short solutions.
How many integration formulas do I need to know for JEE Main?
You need the standard integrals of algebraic, exponential, logarithmic, trigonometric and inverse trigonometric functions, and the special integrals of the form $\dfrac{1}{x^2 \pm a^2}$ and $\sqrt{a^2 \pm x^2}$. The tables in these Indefinite Integral Notes for Class 12 list all of them. Beyond formulas, you need four methods: substitution, by parts, partial fractions and trigonometric identities.
What is the difference between indefinite and definite integrals?
An indefinite integral has no limits and gives a family of functions, $F(x) + C$. A definite integral has upper and lower limits and gives a number. You find the antiderivative first, and then evaluate it at the limits using the fundamental theorem of calculus. eSaral’s Indefinite Integral Notes for Class 12 cover the first step.
How do I choose the method of integration?
First check for a standard form. If the integrand contains a function and its derivative, use substitution. If it is a product of two different types of functions, use integration by parts with the ILATE rule. If it is a rational function, use partial fractions. If it has powers of sine and cosine, use trigonometric identities.
What is the ILATE rule?
ILATE gives the order for choosing $u$ in integration by parts: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Choose $u$ as the function that comes first in this list, and take the other as $v$. For example, in $\displaystyle\int x\,e^x\,dx$, take $u = x$.
Why do we add a constant of integration?
Functions that differ by a constant have the same derivative, so the antiderivative of $f(x)$ is not one function but a family $F(x) + C$. Without $C$, the answer to an indefinite integral is incomplete.
Which topics in indefinite integration are hardest in JEE Advanced?
Integrals that combine several methods, such as a substitution followed by partial fractions, trigonometric integrals using $t = \tan\dfrac{x}{2}$, and reduction formulae, are the usual challenges. Practise them after the standard methods are fast. These Indefinite Integral Notes for Class 12 include the reduction formula for $\int \sin^n x\,dx$.
Which chapter number is Integrals in Class 12 Maths?
In the NCERT Class 12 Mathematics book, Integrals is Chapter 7, and Application of Integrals is Chapter 8. These Indefinite Integral Notes for Class 12 follow the NCERT chapter.
Where can I find more Class 12 Maths notes on eSaral?
eSaral has chapter-wise Class 12 Maths notes, NCERT solutions, the NCERT book and JEE Main chapter-wise PYQs, linked with these Indefinite Integral Notes for Class 12 on this page, so you can study the chapter and practise it in one place.
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eSaral Academic and Editorial Team
Team eSaral is the collective author profile for educational content created by eSaral’s teachers and academic contributors. The team draws on expertise from IIT graduates, doctors, experienced educators and subject specialists to develop resources for JEE, NEET and school students. Our articles aim to explain concepts clearly and help students study with confidence.
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