Integral Calculus Questions for NDA

Indefinite Integration — Standard Forms

Indefinite integration is the reverse of differentiation. When you integrate a function f(x), you find F(x) such that F'(x) = f(x). The result always has a constant C added. Think of it as 'undoing' the derivative. NDA directly tests whether you have memorized the standard integral table — no derivation needed, just recall and apply.

• Always add + C for indefinite integrals — forgetting it is a guaranteed mark loss in theory.
• Integral of x^n = x^(n+1)/(n+1) + C, valid for n not equal to -1.
• Integral of 1/x = ln|x| + C — the absolute value inside log is critical.
• Integral of e^x = e^x + C, integral of a^x = a^x / ln(a) + C.
• Trig integrals: sin x gives -cos x, cos x gives sin x, sec^2 x gives tan x, cosec^2 x gives -cot x.
• Integral of sec x tan x = sec x + C, integral of cosec x cot x = -cosec x + C.

Methods of Integration — Substitution and By Parts

When direct formula does not apply, NDA tests two main techniques. Substitution (t-substitution) works when you spot a function and its derivative both present. Integration by Parts (IBP) is used when two different types of functions are multiplied — like x times e^x, or x times sin x. The ILATE rule tells you which function to choose as the first function for IBP.

• Substitution: if integrand has f(g(x)) times g'(x), substitute t = g(x), dt = g'(x)dx.
• ILATE rule for IBP: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential — pick left-most as first function.
• IBP formula: integral of u.v dx = u times integral(v dx) minus integral(u' times integral(v dx)) dx.
• For integrals like integral of ln x dx, use IBP with u = ln x and v = 1.
• Substitution often used with sqrt expressions — substitute the entire root as t.
• Repeated IBP may be needed for x^2 e^x type problems — reduce power each time.

Definite Integrals and Their Properties

A definite integral gives a numerical value — the area under the curve between two limits. NDA loves testing properties of definite integrals because they let you evaluate tricky integrals without actual computation. These properties are pure formula-based and extremely high yield — memorize all of them.

• Property 1: integral from a to b of f(x)dx = integral from a to b of f(a+b-x)dx — King's property, most used.
• Property 2: integral from 0 to a of f(x)dx = integral from 0 to a of f(a-x)dx — Queen's property.
• Property 3: integral from -a to a of f(x)dx = 2 times integral from 0 to a of f(x)dx if f is even, and 0 if f is odd.
• Property 4: integral from a to b = minus integral from b to a — swapping limits changes sign.
• Property 5: integral from 0 to 2a of f(x)dx = 2 times integral from 0 to a of f(x)dx if f(2a-x)=f(x), else 0.
• Definite integral of a periodic function over full period is same regardless of starting point.

Area Under Curves

NDA asks you to find the area of regions bounded by curves, lines, or between two curves. The area under a curve y = f(x) between x = a and x = b is the definite integral of f(x). If the curve dips below the x-axis, take the absolute value of that integral. Area between two curves is the integral of the top curve minus the bottom curve.

• Area under y=f(x) from a to b = |integral(a to b) f(x)dx| — use modulus if curve is below x-axis.
• Area between two curves = integral(a to b) [f(x) - g(x)]dx where f(x) is upper curve.
• Find intersection points first by solving f(x) = g(x) — these become your limits.
• Area enclosed by a closed curve: split into parts if curve crosses x-axis within limits.
• NDA commonly asks area of region bounded by parabola and a line — standard setup.

• Forgetting the constant C in indefinite integrals — even if the method is perfect, missing C can cost marks in MCQ matching type questions.
• In IBP, choosing the wrong first function — always follow ILATE strictly. Choosing e^x over x as first function makes the problem unsolvable cleanly.
• Not checking even or odd nature before integrating from -a to a — students waste time computing what is simply zero for odd functions.
• Using King's property but forgetting to add both I equations before solving — write I + I = 2I step explicitly, not mentally.
• In area problems, not taking absolute value when curve is below x-axis — integrating a negative area and subtracting it instead of adding gives wrong answer.

• ILATE for Integration by Parts: I Love All The Elephants — Inverse trig, Log, Algebraic, Trig, Exponential.
• King and Queen properties: King replaces x with (a+b-x), Queen replaces x with (a-x) — King rules over both limits, Queen only from 0.
• Even function: graph is Mirror image — integral doubles. Odd function: graph is Rotated — integral cancels to zero. Even = Double, Odd = Zero.
• For trig standard integrals, remember the sign flip: sin integrates to MINUS cos, cos integrates to PLUS sin — opposite to their derivatives.

• NDA typically places 3-4 questions on definite integral properties alone — King's property appears almost every paper. Drill it until it is automatic.
• Questions on area under curve are usually straightforward — parabola vs line or circle vs line. Practice 5-6 standard setups and you will recognize them instantly in exam.
• Time management: indefinite integration questions with substitution should take 60-90 seconds. If you are stuck beyond 2 minutes, move on and return later.
• Standard form integrals (inverse trig, exponential) appear as direct one-step MCQs — these are easy marks. Memorizing the full standard table of 20 integrals is non-negotiable.
• NDA does not ask heavy proof-based questions — focus on application. A question showing an integral and asking which method applies, or giving options for value of a definite integral, are the most common formats.