Indefinite integrals by inspection, substitution, trigonometric identities, standard forms, partial fractions and integration by parts; definite integrals by the fundamental theorem, by substitution and by their properties. Our own step-by-step solutions to Exercises 7.1 to 7.10 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. Read each standard derivative backwards: \(\int x^n\,dx = \dfrac{x^{n+1}}{n+1}\) (\(n \ne -1\)), \(\int\dfrac{dx}{x} = \log|x|\), \(\int e^x\,dx = e^x\), \(\int\cos x\,dx = \sin x\), \(\int\sec^2 x\,dx = \tan x\), \(\int\sec x\tan x\,dx = \sec x\); for \(f(ax + b)\) divide by a. Split sums, expand products and divide out fractions first. Always add C.
What it tests. Look for a function and its derivative in the integrand: put \(t = g(x)\), \(dt = g'(x)\,dx\), integrate in t, then put \(g(x)\) back. Useful results: \(\int\dfrac{f'(x)}{f(x)}\,dx = \log|f(x)|\), \(\int\tan x\,dx = \log|\sec x|\), \(\int\cot x\,dx = \log|\sin x|\).
What it tests. Reduce powers and products to single sines and cosines: \(\sin^2 x = \tfrac{1 - \cos 2x}{2}\), \(\cos^2 x = \tfrac{1 + \cos 2x}{2}\), \(\sin^3 x = \tfrac{3\sin x - \sin 3x}{4}\); \(2\sin A\cos B = \sin(A + B) + \sin(A - B)\), \(2\cos A\cos B = \cos(A + B) + \cos(A - B)\), \(2\sin A\sin B = \cos(A - B) - \cos(A + B)\). Odd powers: keep one factor for dt.
What it tests. Standard forms: \(\int\dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a}\log\left|\dfrac{x - a}{x + a}\right|\), \(\int\dfrac{dx}{a^2 - x^2} = \dfrac{1}{2a}\log\left|\dfrac{a + x}{a - x}\right|\), \(\int\dfrac{dx}{x^2 + a^2} = \dfrac1a\tan^{-1}\dfrac{x}{a}\), \(\int\dfrac{dx}{\sqrt{x^2 \pm a^2}} = \log\left|x + \sqrt{x^2 \pm a^2}\right|\), \(\int\dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac{x}{a}\). Complete the square in a quadratic; for \(\dfrac{px + q}{\text{quadratic}}\) write \(px + q = A\,(\text{quadratic})' + B\).
What it tests. Make the fraction proper first (divide if needed). Then \(\dfrac{px + q}{(x - a)(x - b)} = \dfrac{A}{x - a} + \dfrac{B}{x - b}\), a repeated factor \((x - a)^2\) gives \(\dfrac{A}{x - a} + \dfrac{B}{(x - a)^2}\), and an irreducible quadratic gives \(\dfrac{Bx + C}{x^2 + bx + c}\). Find constants by putting convenient x values (cover-up) and comparing coefficients.
What it tests. \(\int u\,v\,dx = u\int v\,dx - \int\left(u'\int v\,dx\right)dx\); choose u by ILATE (inverse trig, log, algebraic, trig, exponential). Useful result: \(\int e^x[f(x) + f'(x)]\,dx = e^xf(x)\). For \(\int e^{ax}\sin bx\,dx\), apply parts twice and solve for the integral.
What it tests. \(\int\sqrt{a^2 - x^2}\,dx = \dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}\), \(\int\sqrt{x^2 \pm a^2}\,dx = \dfrac{x}{2}\sqrt{x^2 \pm a^2} \pm \dfrac{a^2}{2}\log\left|x + \sqrt{x^2 \pm a^2}\right|\). Complete the square first and shift x.
What it tests. Second fundamental theorem: if \(F' = f\) on \([a, b]\), then \(\int_a^b f(x)\,dx = F(b) - F(a)\); no C is needed. Find an antiderivative by any method, then substitute the limits carefully (brackets!).
What it tests. After substituting \(t = g(x)\), change the limits too (\(x = a \to t = g(a)\)) so you never need to go back to x.
What it tests. \(\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx\) (add the two forms of I); \(\int_{-a}^a f = 2\int_0^a f\) for even f and 0 for odd f; \(\int_0^{2a} f = 2\int_0^a f\) if \(f(2a - x) = f(x)\), 0 if \(f(2a - x) = -f(x)\); split at the zeros of \(|\,\cdot\,|\) expressions.
What it tests. Mixed practice: pick the method from the shape of the integrand (substitution, partial fractions, parts, standard forms), simplify algebraically or trigonometrically first, and for definite integrals try the properties before brute force.
Integrals has 43 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Integrals · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.