What the board asks

We went through the ISC board papers, question by question, and matched each question to a topic. Here is what was asked, how often, and what was never asked.

Mathematics: 309 topics in the 2027 syllabus, across 11 units. The board has asked 255 of them at least once. 54 were never asked in the papers we checked.

Pick a unit

M5 Calculus: Integrals

All the topics in Mathematics: Calculus: Integrals

Asked topics first, the most asked at the top. Tap a topic to see what to write, the result, the years, and the real board questions.

53 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Definite integral of a modulus functionFind where the expression inside \|.\| is zero, split the interval there, write each piece without the modulus with the correct sign, integrate and add.	Sum	1	5 papers	2026	Asked
2	Property f(a-x) or f(a+b-x): add the integral to itselfWrite I, apply the property to get a second form of I, add the two so the integrand simplifies (often to 1 or a standard form), evaluate 2I and halve.	Sum	3	5 papers	2026	Asked
3	Substitution: f'(x)[f(x)]^n and f'(x) g(f(x))Choose t = f(x), write dt = f'(x) dx, rewrite the whole integral in t, integrate, substitute back and add + C.	Sum	2	3 papers	Specimen 2027	Asked
4	By parts with a single log or inverse-trig functionTake the function as u and 1 as v, apply the by-parts formula, simplify the new integral (by substitution if it is not standard) and add + C.	Sum	2	3 papers	Specimen 2027	Asked
5	Evaluate a definite integral with the fundamental theoremFind an anti-derivative F, write [F(x)] from a to b, compute F(b) - F(a) exactly and simplify (no + C).	Sum	1	3 papers	Specimen 2027	Asked
6	Odd and even functions on -a to aTest f(-x) against f(x), use 0 for an odd function or 2 times the integral from 0 to a for an even one (split a sum into its odd and even parts first), then evaluate.	Sum	1	3 papers	Specimen 2027	Asked
7	Substitution (e^x, sin x, cos x) then partial fractionsPut t = e^x or sin x or cos x so the integrand becomes a rational function of t, split it into partial fractions, integrate, substitute back and add + C.	Sum	5	3 papers	2026	Asked
8	Substitution that reduces to a quadratic standard formSpot the inner function (sin x, e^x, x^2), put t equal to it so the integral becomes dt/(t^2 + bt + c) or similar, apply the standard form, substitute back and add + C.	Sum	2	3 papers	2026	Asked
9	Trig integrand rewritten by identities into standard formsUse an identity (tan^2 = sec^2 - 1, split a fraction into sec/cosec products, add and subtract a term) to turn the integrand into a sum of standard derivatives, then integrate and add + C.	Sum	2	3 papers	2025 improvement	Asked
10	x times f(trig): property removes x, then evaluateApply the f(a + b - x) or f(a - x) property so x becomes (a + b - x), add to get (a + b)/2 times an integral without x, then evaluate that integral by a trig identity or standard form.	Proof	5	3 papers	2025 improvement	Asked
11	Six standard results: 1/(x^2 +- a^2), 1/sqrt(x^2 +- a^2) etc.State each standard result with the correct constant and sign and use it directly, adding + C.	Statement	1	2 papers	Specimen 2027	Asked
12	Integrals of 1/(a cos^2 x+b sin^2 x+c)Divide numerator and denominator by cos^2 x, write the denominator in tan x, put t = tan x so dt = sec^2 x dx, apply the quadratic standard form and add + C.	Sum	3	2 papers	2026	Asked
13	Integrate after dividing, expanding or splitting the integrandDivide each term by the denominator or expand the product, simplify to a sum of powers and standard functions, integrate term by term and add + C.	Sum	2	2 papers	2025	Asked
14	Find the constant K in a given anti-derivativeSubstitute t = the inner function, integrate to get the answer form, and compare with the given expression (or differentiate the given answer) to read off K.	Sum	2	2 papers	2025 improvement	Asked
15	By parts: polynomial times log or inverse-trig functionTake the log or inverse-trig function as u and the polynomial as v, apply by parts, simplify the rational integral left (divide or add and subtract) and add + C.	Sum	3	2 papers	2025 improvement	Asked
16	Partial fractions: linear and irreducible quadratic factorWrite A/(x-a) + (Bx + C)/(x^2 + k), find A from the root and B, C by comparing coefficients, integrate the quadratic part as a log term plus a tan^-1 term, and add + C.	Sum	5	2 papers	2025 improvement	Asked
17	Property for 0 to 2a: f(2a-x) = +-f(x)Check whether f(2a - x) equals f(x) or -f(x), write 2 times the integral from 0 to a or 0, then evaluate.	Sum	3	2 papers	2025 improvement	Asked
18	Integrals of (px + q)/(ax^2 + bx + c)Write px + q = A(2ax + b) + B, find A and B, integrate the first part as a log (f'/f) and the second by completing the square, add + C.	Sum	3	2 papers	2024	Asked
19	Integrals of the form f'(x)/f(x)Recognise the numerator as (a multiple of) the derivative of the denominator, adjust the constant, write log\|f(x)\| + C.	Sum	1	2 papers	2023	Asked
20	Simplify an inverse-trig integrand before integratingRewrite the inverse-trig expression (by an identity, or a substitution such as x = cos 2t) into a simple function of x or t, then integrate, by parts if needed, and add + C.	Sum	3	2 papers	2023	Asked
21	Compare two answers: anti-derivatives differ by a constantDifferentiate each given answer (or rewrite one with an identity) to show both have the same derivative, so they differ only by a constant, and choose the option.	Give reason	1	1 paper	Specimen 2027	Asked
22	Find f(x) from f'(x) and a value; spot errorsIntegrate f'(x), keep + C, put in the given point to find C, write f(x); when a worked solution is shown, check each step and name the wrong one or say none.	Give reason	1	1 paper	Specimen 2027	Asked
23	Integrals of tan x, cot x, sec x, cosec xWrite tan x = sin x / cos x (or multiply sec x by (sec x + tan x)/(sec x + tan x)), use the f'/f form, and write the log result with + C.	Sum	2	1 paper	Specimen 2027	Asked
24	Two-part 'hence' question: reuse part (a) after substitutionUse a substitution or an identity (for example tan(A+B) for a product of tangents) to bring the new integral to the form found in part (a), quote that result, substitute back and add + C.	Sum	3	1 paper	Specimen 2027	Asked
25	Basic properties: reverse limits, dummy variable, split at cUse the stated property by name (reverse the limits with a minus, change the dummy variable, or join or split the interval at c) to rewrite and evaluate from given values.	Sum	2	1 paper	Specimen 2027	Asked
26	Assertion-reason or two statements on integral propertiesCheck the Assertion by the right property, check the Reason is true as a statement, decide whether the Reason is the property actually used, and choose the option.	Give reason	1	1 paper	Specimen 2027	Asked
27	Linear substitution: f(ax + b)Integrate as if the variable were ax + b and divide by a (or put t = ax + b and change dx), then add + C.	Sum	1	1 paper	2025	Asked
28	Partial fractions: repeated linear and quadratic factorsWrite A/(x-a) + B/(x-a)^2 + (Cx + D)/(x^2 + k), find all four constants, integrate each part and add + C.	Sum	5	1 paper	2025	Asked
29	Integrals of 1/(a+b cos x) or 1/(a+b sin x)Put t = tan(x/2), write cos x or sin x and dx in t, reduce to dt/(quadratic), apply the standard form, put back tan(x/2) and add + C.	Sum	3	1 paper	2025	Asked
30	By parts: polynomial times sin, cos or e^xTake the polynomial as u (ILATE), apply by parts repeatedly until the polynomial is gone, tidy the signs and add + C.	Sum	3	1 paper	2024	Asked
31	Integrals of the form e^x [f(x) + f'(x)]Split the bracket into a function f and its derivative f', write the answer e^x f(x) + C (justify by parts or by differentiating).	Sum	1	1 paper	2024	Asked
32	Standard anti-derivatives of basic functionsWrite the anti-derivative of each basic function from memory, check one by differentiating, and add + C.	Statement	1	1 paper	2023	Asked
33	sin^m x cos^n x with an odd power, by substitutionSplit off one factor of the odd power, change the rest with sin^2 + cos^2 = 1, put t = cos x or sin x, integrate the polynomial in t and substitute back, adding + C.	Sum	3	1 paper	2020	Asked
34	Partial fractions: distinct linear factorsFactorise the denominator, write A/(x-a) + B/(x-b) (+ C/(x-c)), find the constants by putting x = roots, integrate each to a log and add + C.	Sum	3	1 paper	2020	Asked
35	Integrals of 1/sqrt(ax^2+bx+c) by completing the squareTake sqrt(\|a\|) out, complete the square, recognise sin^-1 or log form, write it with + C.	Sum	3	1 paper	2020	Asked
36	Integrals of (px + q)/sqrt(ax^2 + bx + c)Write px + q = A(2ax + b) + B, integrate the first part as 2A sqrt(quadratic) and the second by completing the square, add + C.	Sum	3	1 paper	2018	Asked
37	Integrals of sin^2 x, cos^2 x, sin^4 x, cos^4 xWrite the power-reduction identity (twice for the 4th power), expand into constants and cosines of multiple angles, integrate each term and add + C.	Sum	2	never		Never asked
38	Integrals of sin^3 x and cos^3 xUse the triple-angle identity (or split off one factor and substitute), integrate each term and add + C.	Sum	2	never		Never asked
39	Cyclic by parts: e^(ax) sin bx or e^(ax) cos bxApply by parts twice keeping e^(ax) as the same choice, bring the repeated integral I to the left side, solve for I and add + C.	Sum	3	never		Never asked
40	Partial fractions: repeated linear factorWrite A/(x-a) + B/(x-a)^2 + C/(x-b), find the constants by roots and comparing coefficients, integrate (the squared term gives -1/(x-a)) and add + C.	Sum	3	never		Never asked
41	Improper rational function: divide first, then partial fractionsWhen degree of numerator >= degree of denominator, divide to get a polynomial plus a proper fraction, split the proper fraction into partial fractions, integrate and add + C.	Sum	3	never		Never asked
42	Integrals of 1/(ax^2+bx+c) by completing the squareTake a out, complete the square to get (x + p)^2 +- k^2, pick the matching standard result, write it with + C.	Sum	2	never		Never asked
43	Integrals of sqrt(a^2-x^2), sqrt(x^2+a^2), sqrt(x^2-a^2)Quote (or derive by parts) the standard result, substitute the values of a and x, and add + C.	Sum	2	never		Never asked
44	Integrals of sqrt(ax^2 + bx + c)Complete the square to reach one of the three sqrt forms, apply the standard result with the shifted variable, add + C.	Sum	3	never		Never asked
45	Integrals of (px + q) sqrt(ax^2 + bx + c)Write px + q = A(2ax + b) + B, integrate A f' sqrt(f) as (2A/3) f^(3/2) and B sqrt(quadratic) by the standard form, add + C.	Sum	5	never		Never asked
46	Integrals of 1/(a cos x + b sin x)Write a = r sin alpha, b = r cos alpha (or the cos form) so the denominator is r sin(x + alpha), integrate cosec, and add + C.	Sum	3	never		Never asked
47	Integrals of 1/(a cos x+b sin x+c)Put t = tan(x/2), replace sin x, cos x and dx, collect the quadratic in t, complete the square, apply the standard form and add + C.	Sum	3	never		Never asked
48	Ratio (a cos x+b sin x)/(c cos x+d sin x)Write numerator = A(denominator) + B(derivative of denominator), find A and B by comparing coefficients, integrate to Ax + B log\|denominator\| and add + C.	Sum	3	never		Never asked
49	Integrals of (1 +- x^2)/(1 + x^4)Divide top and bottom by x^2, write the denominator as (x -+ 1/x)^2 +- 2, put t = x -+ 1/x, apply the standard form and add + C.	Sum	3	never		Never asked
50	Integrals of 1/(1 + x^4) and x^2/(1 + x^4)Write the integrand as half the sum or difference of (1 + x^2)/(1 + x^4) and (1 - x^2)/(1 + x^4), integrate each by MA5-41 and combine with + C.	Sum	5	never		Never asked
51	Integrals of sqrt(tan x) and sqrt(cot x)Put tan x = t^2 so dx = 2t dt/(1 + t^4), reduce to an integral of t^2/(1 + t^4) type, solve by MA5-41, substitute back and add + C.	Sum	5	never		Never asked
52	Definite integral by substitution or by parts, with limitsMake the substitution and change the limits to the new variable (or apply by parts with [uv] from a to b), evaluate and simplify.	Sum	3	never		Never asked
53	Standard result: integral 0 to pi/2 of log sin xUse the f(a - x) property to get the cos form, add, use sin x cos x = (sin 2x)/2, substitute 2x = t and the 0 to 2a property to solve for I.	Sum	5	never		Never asked

This list was made from the 2027 syllabus and checked with 10 board papers, including the 2027 specimen paper. It is not yet checked with the textbook.

The 10 topics Mathematics papers ask most

Counted on the board papers from 2018 to the 2027 specimen paper, which every unit has been checked against.

1.Formulate an LPP from a word problemM10 Linear Programming · asked 7 times since 2018
2.Scalar and vector projection of one vector on anotherM8 Vector Algebra · asked 6 times since 2018
3.Mean (expected value) of a random variableM11 Probability · asked 6 times since 2018
4.Find the inverse of an invertible algebraic function and verifyM1 Relations and Functions · asked 5 times since 2018
5.Principal value of expressions and nested inverse trigM1 Relations and Functions · asked 5 times since 2018
6.Evaluate or show zero without expanding, using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
7.Prove a determinant equals a factorised expression using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
8.Definite integral of a modulus functionM5 Calculus: Integrals · asked 5 times since 2018
9.Case study or linked parts on lines and planesM9 Three-Dimensional Geometry · asked 5 times since 2018
10.Equations solved with the complementary identitiesM1 Relations and Functions · asked 4 times since 2018

See when each unit comes in your plan