Class 12 Maths NCERT Solutions Chapter 7 Integrals | Boundless Maths

Unit III · Calculus · Chapter 7

Class 12 Maths NCERT Solutions Chapter 7: Integrals

Free, step-by-step NCERT Solutions for all ten exercises of this chapter — anti derivatives by inspection, integration by substitution, trigonometric identities, the standard integrals of particular functions, partial fractions, integration by parts, special integral forms, definite integrals, evaluating definite integrals by substitution, and the properties that make definite integrals easier to solve. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

Integrals is the biggest chapter in the entire Class 12 Maths syllabus — ten exercises and more marks riding on it than any other single chapter in Calculus. It's also the most varied: instead of one core technique, you're really learning six overlapping toolkits, and the real skill CBSE tests is recognising which one a given question needs.

📐 Formula Deck 🤖 AI Question Bank

11Exercises (incl. Misc.)

261Total Questions

2026-27CBSE Syllabus

100%Solved

Chapter Overview

Integrals reverses everything Chapters 5 and 6 built: instead of differentiating a function, you're given its derivative and asked to find the original function back — the anti derivative, or indefinite integral. Because differentiation isn't one-to-one (many functions share the same derivative up to a constant), every indefinite integral carries an arbitrary constant of integration, C, and the family of anti derivatives {F(x) + C} is what you're really finding when you integrate.

The chapter then builds an entire toolkit for finding these anti derivatives when inspection alone isn't enough: substitution (recognising a function and its derivative together in the same expression), trigonometric identities (rewriting products and powers of sin, cos and tan as sums you can integrate directly), the six standard forms for expressions like 1/(x²−a²) and 1/√(a²−x²), partial fractions (breaking a rational function into simpler pieces), and integration by parts (for products of unlike functions like x·eˣ or x·log x). The chapter closes with definite integrals, where the Fundamental Theorem of Calculus turns integration into a simple substitution of limits, and a set of properties that can turn an intimidating definite integral into a single line of algebra.

How the Chapter Builds

One Idea Leads to the Next

Anti Derivatives

Reverse a standard derivative formula to find F such that F′(x) = f(x). Exercise 7.1.

→

Substitution & Trig Identities

Change of variable when a derivative hides in the integrand; rewrite trig products as sums. Exercises 7.2–7.3.

→

Particular Functions & Partial Fractions

Six standard forms after completing the square; decompose rational functions into simpler pieces. Exercises 7.4–7.5.

→

By Parts & Special Forms

∫u dv = uv − ∫v du for products of unlike functions; special forms like √(x²−a²). Exercises 7.6–7.7.

→

Definite Integrals & FTC

Evaluate using F(b) − F(a); substitution with limits changed to the new variable. Exercises 7.8–7.9.

→

Properties of Definite Integrals

Symmetry and interval-manipulation shortcuts (P₀–P₇) that avoid direct integration entirely. Exercise 7.10.

Quick Reference

Important Formulas — Chapter 7

Everything you need before you start solving. This is a summary for quick recall — the Formula Deck below has the full printable version for all of Calculus.

Standard Integrals (§7.1–7.2)

Power rule

\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C,\; n\neq -1

Particularly, \int dx = x + C.

Trigonometric integrals

\int \cos x\,dx = \sin x + C,\quad \int \sin x\,dx = -\cos x + C

\int \sec^2 x\,dx = \tan x + C,\quad \int \text{cosec}^2 x\,dx = -\cot x + C

The building blocks for every trig-based integral in this chapter.

Exponential & logarithmic

\int e^x\,dx = e^x + C,\quad \int a^x\,dx = \dfrac{a^x}{\log a} + C,\quad \int \dfrac{1}{x}\,dx = \log|x| + C

Never drop the modulus in the log formula — the domain includes negative x too.

Integrals of Particular Functions (§7.4)

Difference of squares

\int \dfrac{dx}{x^2-a^2} = \dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right| + C

Complete the square first if the denominator isn't already in this form.

Sum of squares

\int \dfrac{dx}{x^2+a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + C

Watch for this form hiding inside ax² + bx + c after completing the square.

Square root of a difference

\int \dfrac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\dfrac{x}{a} + C

Three more forms of this family appear in the Formula Deck — all six follow the same completing-the-square approach.

Integration by Parts (§7.6)

Integration by parts

\int u\,v\,dx = u\int v\,dx - \int\left[\dfrac{du}{dx}\int v\,dx\right]dx

Choose the first function u using ILATE — Inverse, Log, Algebraic, Trig, Exponential — in that priority order.

The eˣ[f(x) + f′(x)] shortcut

\int e^x\left[f(x)+f'(x)\right]dx = e^x f(x) + C

Spotting this pattern skips integration by parts entirely.

Definite Integrals & Key Properties (§7.8, §7.10)

Fundamental Theorem of Calculus

\int_a^b f(x)\,dx = F(b) - F(a),\quad \text{where } F'(x) = f(x)

No need to keep the constant C — it cancels out between F(b) and F(a).

Property: a + b − x substitution

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

Often lets you add two copies of the same integral to cancel out the hard part.

Property: even and odd functions

If f is even, \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx; if f is odd, the integral is 0.

Check f(−x) before integrating — it can turn a hard problem into a one-liner.

Decision Guide

Which Integration Method Should I Use?

A quick way to decide, once you've looked at the integrand.

What the integrand looks like	Use this	Why
A function's derivative also appears in the expression (e.g. 2x·sin(x²+1))	Substitution	Reduces the integral to ∫f(t) dt directly (§7.2).
A product or power of sin, cos or tan (e.g. sin³x·cos²x)	Trigonometric identities	Converts the product into a sum of terms you can integrate directly (§7.3).
Matches 1/(x²±a²), 1/√(a²−x²) etc., or has ax²+bx+c in the denominator	Particular function formulas	Complete the square first, then apply one of the six standard forms (§7.4).
A proper rational function with a factorisable denominator	Partial fractions	Breaks the fraction into simpler pieces that resolve to log or tan⁻¹ forms (§7.5).
A product of two unlike functions (x·eˣ, x·log x, x·sin x)	Integration by parts (ILATE order)	The wrong choice of first function makes the integral harder, not easier (§7.6).
Contains √(x²−a²), √(x²+a²) or √(a²−x²)	Special integral forms	These follow a fixed integration-by-parts pattern worth memorising (§7.7).
A definite integral with symmetric limits (0 to a, −a to a) or limits that add neatly	Properties of definite integrals	P₀–P₇ can solve it in one line, before you ever integrate directly (§7.10).

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all ten exercises.

Forgetting the constant of integration, C, on indefinite integrals — CBSE deducts a mark for this every single time, even when the rest of the working is correct.
Forgetting to change the limits when substituting in a definite integral — the new limits must be worked out in terms of the new variable, not left as the old x-limits.
Sign errors in inverse trigonometric standard forms — mixing up which form uses sin⁻¹x versus −cos⁻¹x, or dropping the negative sign in ∫dx/√(1−x²) = −cos⁻¹x + C.
Wrong choice of "first function" in integration by parts — not following ILATE (Inverse, Log, Algebraic, Trig, Exponential) order leads to a harder integral, not an easier one.
Dropping the modulus in log-based integrals — writing log x instead of log|x|, which silently restricts the domain to x > 0 only.
Attempting to integrate sin³x or cos⁴x directly instead of rewriting it first using a trigonometric identity like sin 3x = 3 sin x − 4 sin³x.
Missing partial fraction terms for repeated or quadratic factors — a repeated linear factor (x−a)² needs both A/(x−a) and B/(x−a)², and an irreducible quadratic factor needs (Bx+C)/(x²+bx+c), not just a constant.
Applying a definite integral property without checking its condition first — for example, using P₆ (∫₀^2a f(x)dx = 2∫₀^a f(x)dx) without first verifying that f(2a−x) = f(x).

Solve Chapter-Wise

Choose an Exercise

7.1

Exercise 7.1

Finding anti derivatives by inspection — reversing standard derivative formulas · 22 questions

Solve Exercise 7.1 →

7.2

Exercise 7.2

Integration by substitution — trigonometric, exponential and logarithmic integrands · 39 questions

Solve Exercise 7.2 →

7.3

Exercise 7.3

Integrals using trigonometric identities — products and powers of sin, cos and tan · 24 questions

Solve Exercise 7.3 →

7.4

Exercise 7.4

Standard integrals of particular functions — 1/(x²±a²), 1/√(a²−x²) and related forms · 25 questions

Solve Exercise 7.4 →

7.5

Exercise 7.5

Integration by partial fractions — proper and improper rational functions · 23 questions

Solve Exercise 7.5 →

7.6

Exercise 7.6

Integration by parts — products of algebraic, trigonometric, exponential and log functions · 24 questions

Solve Exercise 7.6 →

7.7

Exercise 7.7

Special integral forms — √(x²−a²), √(x²+a²) and √(a²−x²) via integration by parts · 11 questions

Solve Exercise 7.7 →

7.8

Exercise 7.8

Evaluating definite integrals using the Fundamental Theorem of Calculus · 22 questions

Solve Exercise 7.8 →

7.9

Exercise 7.9

Evaluating definite integrals by substitution, with limits changed to the new variable · 10 questions

Solve Exercise 7.9 →

7.10

Exercise 7.10

Properties of definite integrals — shortcuts using symmetry and interval manipulation · 21 questions

Solve Exercise 7.10 →

Miscellaneous Exercise

Mixed integration problems combining every technique from the chapter, plus proofs · 40 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Calculus — differentiation, applications of derivatives, integration — in one printable PDF.

Get Formula Deck →

🤖 Want Board Exam-Style Practice?

The AI Question Bank has board-tagged MCQs, Assertion-Reason and Case Studies for Integrals, with instant feedback.

Explore AI Q-Bank →

Common Questions

Frequently Asked Questions

Quick answers about Chapter 7, Integrals.

How many exercises are there in Chapter 7, Integrals?

There are ten main exercises — 7.1 through 7.10 — plus a Miscellaneous Exercise, totalling 261 questions covering anti derivatives, substitution, trigonometric identities, standard integral forms, partial fractions, integration by parts, special integrals, definite integrals, and their properties. It is the largest chapter in the Class 12 Maths syllabus.

Is this chapter important for the board exam?

Yes — it's the single largest chapter in Class 12 Maths and carries the highest weightage within the Calculus unit. Questions from Integrals appear across nearly every section of the board paper, from 1-mark MCQs to 5-mark long-answer questions.

Which integration method should I use?

It depends on the integrand: use inspection or standard formulas for simple forms, substitution when a function's derivative also appears in the integrand, trigonometric identities for products or powers of trig functions, partial fractions for rational functions, and integration by parts for products of unlike functions such as x·eˣ. See the decision guide above for the full breakdown.

What should I revise before starting this chapter?

Make sure Chapter 5 (Continuity and Differentiability) is solid, since every integration technique is really the reverse of a differentiation rule — the chain rule becomes substitution, and product-rule differentiation becomes integration by parts.

Where can I find the official NCERT textbook for this chapter?

Integrals is Chapter 7 of the NCERT Class 12 Mathematics textbook (Part II), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.