Class 12 Maths · Chapter 7
Unit: Calculus
Tentative Study Share: 7.0 Marks
Chapter 7: Integrals — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Integrals.
NCERT Exercises in Chapter 7: Integrals
Exercise 7.1
22 NCERT Questions Solved
Exercise 7.239 NCERT Questions Solved
Exercise 7.324 NCERT Questions Solved
Exercise 7.425 NCERT Questions Solved
Exercise 7.523 NCERT Questions Solved
Exercise 7.624 NCERT Questions Solved
Exercise 7.711 NCERT Questions Solved
Exercise 7.822 NCERT Questions Solved
Exercise 7.910 NCERT Questions Solved
Exercise 7.1021 NCERT Questions Solved
Miscellaneous Exercise40 NCERT Questions Solved
📘 NCERT Concept Notes — Integrals
01 · Integration reverses differentiation
An indefinite integral is a family of antiderivatives, differing by a constant on a domain interval. Check your answer by differentiating it. Keep the original real domain even when algebra cancels a denominator.
02 · Choose a method from the structure
Look for an inner derivative for substitution, simplify trigonometric powers using identities, divide improper rational functions before partial fractions, and use parts when differentiating one factor simplifies the product.
03 · Recognise the standard denominator
Complete the square to decide which standard integral applies. A numerator that is nearly the denominator derivative can be split into a derivative part and a remainder.
04 · Definite integrals use limits, not arbitrary constants
Evaluate a primitive at the upper endpoint minus the lower endpoint. Under substitution, change both limits. Symmetry is useful only when the integral exists; cancellation across divergent endpoints does not establish convergence.
⚡ Key Formulas & Results — Integrals
Power and logarithm
Which exponent is the exception to the power rule?
\(\int x^n dx=\frac{x^{n+1}}{n+1}+C\ (n\ne-1),\quad\int\frac{dx}{x}=\log|x|+C\)
Linearity of integration
How do sums and constant factors behave?
\(\int(af+bg)dx=a\int fdx+b\int gdx\)
Exponential integrals
Which constants divide exponential antiderivatives?
\(\int e^{kx}dx=\frac{e^{kx}}k+C,\quad\int a^x dx=\frac{a^x}{\log a}+C\)
Basic trigonometric integrals
Recall the sine, cosine, secant-square and cosecant-square integrals.
\(\begin{aligned}\int\sin xdx&=-\cos x+C\\\int\cos xdx&=\sin x+C\\\int\sec^2x dx&=\tan x+C\\\int\operatorname{cosec}^2x dx&=-\cot x+C\end{aligned}\)
Secant and cosecant products
Which products directly reverse familiar derivatives?
\(\int\sec x\tan xdx=\sec x+C,\quad\int\operatorname{cosec}x\cot xdx=-\operatorname{cosec}x+C\)
Substitution
How does u=g(x) transform an integral?
\(\int f(g(x))g\prime(x)dx=\int f(u)du\)
Trigonometric logarithms
Recall integrals of tan, cot, sec and cosec.
\(\begin{aligned}\int\tan xdx&=-\log|\cos x|+C\\\int\cot xdx&=\log|\sin x|+C\\\int\sec xdx&=\log|\sec x+\tan x|+C\\\int\operatorname{cosec}x dx&=\log|\operatorname{cosec}x-\cot x|+C\end{aligned}\)
Trigonometric powers
Which identities reduce squared sine and cosine?
\(\sin^2x=\frac{1-\cos2x}{2},\quad\cos^2x=\frac{1+\cos2x}{2}\)
Partial-fraction shapes
What numerator belongs above an irreducible quadratic factor?
\(\frac{P(x)}{(x-a)(x^2+bx+c)}=\frac A{x-a}+\frac{Bx+C}{x^2+bx+c}\)
Derive integration by parts
Which differentiation rule produces the parts formula?
\(\int u\,dv=uv-\int v\,du\)
🎯 CBSE 2026 Board PYQs from Integrals
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q8
If ∫ (3ax)/(b² + c²x²) dx = A log| b² + c²x² | + K, then the value of A is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q9
The value of ∫₋₁¹ x³/(x² + 2|x| + 1) dx is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q26
Evaluate: ∫₀¹ x tan⁻¹ x dx.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q27
Find: ∫ √((x+2)/(x−2)) dx. OR Find: ∫ x²/((x²+9) (x²+16)) dx.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q28
If I₁ = ∫₋π/4^π/4 dx/(1+cos2x) and I₂ = ∫₋1/2^1/2 |x| dx, then show that I₁ − 4I₂ = 0.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q1
If ∫ (3ax)/(b² + c²x²) dx = A log| b² + c²x² | + K, then the value of A is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q2
The value of ∫₋₁¹ x³/(x² + 2|x| + 1) dx is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q26
If I₁ = ∫₋π/4^π/4 dx/(1+cos2x) and I₂ = ∫₋1/2^1/2 |x| dx, then show that I₁ − 4I₂ = 0.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q30
Evaluate \(\int_0^1 x\sin^{-1}x\,dx\).
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q31
Find: ∫ √((x+2)/(x−2)) dx. OR Find: ∫ x²/((x²+9) (x²+16)) dx.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q12
If ∫ (3ax)/(b² + c²x²) dx = A log| b² + c²x² | + K, then the value of A is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q13
The value of ∫₋₁¹ x³/(x² + 2|x| + 1) dx is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q26
Evaluate \(\int_0^1 \ln(1+x^2)\,dx\).
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q28
Find: ∫ √((x+2)/(x−2)) dx. OR Find: ∫ x²/((x²+9) (x²+16)) dx.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q29
If I₁ = ∫₋π/4^π/4 dx/(1+cos2x) and I₂ = ∫₋1/2^1/2 |x| dx, then show that I₁ − 4I₂ = 0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q9
If \(\int_0^{2a}\frac{dx}{1+4x^2}=\frac{\pi}{6}\), find a.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q26
Find \(\int\frac{x+2}{\sqrt{9x-x^2}}\,dx\).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q27
Evaluate \(\int_{\pi/12}^{5\pi/12}\frac{dx}{1+\sqrt{\cot x}}\). OR Evaluate…
📝 Chapterwise Practice Questions — Integrals (177 Questions)
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).
Class 12 Integrals · Mcq Single Correct
For real a,b with \(\sin(b-a)\ne0\), evaluate \(\int dx/[\sin(x-a)\sin(x-b)]\) on an interval avoiding…
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int\arctan\sqrt{x}\,dx\) for x>0.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x[(1-x)/(1+x^2)]^2\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int2^{x+2}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-\pi/4}^{\pi/4}\sec^2x\,dx\).
Class 12 Integrals · Mcq Single Correct
For x≠0, evaluate \(\int x^9/(4x^2+1)^6\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_0^{\pi/8}\tan^2(2x)\,dx\).
Class 12 Integrals · Mcq Single Correct
For continuous f on the relevant intervals, \(\int_{a+c}^{b+c}f(x)dx\) equals:
Class 12 Integrals · Mcq Single Correct
Let a>0 and f,g be continuous on [0,a], with f(x)=f(a-x) and g(x)+g(a-x)=a. Find \(I=\int_0^a…
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}(x^3+|x|+1)/(x^2+2|x|+1)\,dx\).
Class 12 Integrals · Mcq Single Correct
If \(\int_0^1e^t/(1+t)dt=a\), find \(\int_0^1e^t/(1+t)^2dt\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-2}^{2}|x\cos(\pi x)|\,dx\).
Class 12 Integrals · Mcq Single Correct
If f′(x)=x+1/x for x≠0, find f(x) on a connected domain interval.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_2^3 x/(x^2+1)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_0^2\sqrt{4-x^2}\,dx\).
Class 12 Integrals · Mcq Single Correct
If f′(x)=ln x for x>0, find f(x).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_0^{\pi/6}\sec^2(x-\pi/6)\,dx\).
Class 12 Integrals · Mcq Single Correct
If f′(x)=ax+b and f(0)=0, find f(x).