Basic Integration and Definite Integrals Questions for AGNIVEER NAVY

Indefinite Integration Basics

Integration is the reverse of differentiation. When you integrate a function, you find its antiderivative — the function whose derivative gives you the original. The indefinite integral includes a constant C because many functions have the same derivative. Think of it like reversing a recipe: if differentiation breaks down a cake into ingredients, integration reassembles them. The key is memorizing standard integral formulas and recognizing which one applies.

• Indefinite integral of f(x) is written as ∫f(x)dx = F(x) + C, where F'(x) = f(x)
• Always add constant C — this is non-negotiable in indefinite integrals
• Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1
• ∫1/x dx = ln|x| + C (absolute value is critical for negative x)
• Linearity: ∫[af(x) + bg(x)]dx = a∫f(x)dx + b∫g(x)dx
• Integration is additive — split complex expressions into simpler parts

Trigonometric Integration

Trig integrals appear frequently in Agniveer Navy papers. The key formulas are straightforward — just memorize the sign changes. Sine integrates to negative cosine, cosine integrates to positive sine. A useful memory trick: think of the derivative chain: d/dx(sin x) = cos x, so ∫cos x dx = sin x. The reverse applies for cosine.

• ∫sin(x) dx = -cos(x) + C (negative sign is critical)
• ∫cos(x) dx = sin(x) + C
• ∫sec^2(x) dx = tan(x) + C
• ∫cosec^2(x) dx = -cot(x) + C
• ∫sec(x)tan(x) dx = sec(x) + C
• ∫cosec(x)cot(x) dx = -cosec(x) + C

Definite Integrals and Fundamental Theorem

A definite integral has upper and lower limits and gives a numerical answer (no constant C). The Fundamental Theorem of Calculus links integration and differentiation: to evaluate a definite integral, find the antiderivative F(x), then compute F(b) - F(a) where a and b are the limits. This is the core technique for all definite integral problems in Agniveer Navy exams.

• Definite integral notation: ∫[a to b] f(x) dx = F(b) - F(a), where F is the antiderivative
• Always evaluate: substitute upper limit minus lower limit
• Limits of integration matter — swapping them changes the sign: ∫[a to b] = -∫[b to a]
• If upper and lower limits are equal, the integral is zero: ∫[a to a] f(x) dx = 0
• Definite integrals can be split: ∫[a to c] f(x) dx = ∫[a to b] f(x) dx + ∫[b to c] f(x) dx
• Geometrically, definite integral represents the signed area under the curve

Integration by Substitution (u-substitution)

Substitution is used when the integrand is a composite function. The idea: replace a complex expression with a simpler variable u, find du, and rewrite the integral in terms of u. After integration, substitute back. This is moderately tested in Agniveer Navy — expect 0-1 question per paper. The key is recognizing when to use it: look for a function and its derivative present together.

• Choose u as the inner function (the one being composed)
• Find du/dx, then express dx in terms of du
• Rewrite the entire integral in terms of u and du
• Integrate with respect to u
• Substitute back the original variable
• For definite integrals, change the limits when you substitute u

• Forgetting the constant C in indefinite integrals — this costs marks. Every indefinite integral must end with + C
• Mixing up ∫sin(x) dx = -cos(x) with ∫cos(x) dx = sin(x) — the negative sign on sine is easy to drop
• Evaluating definite integrals backwards: writing F(a) - F(b) instead of F(b) - F(a) — this flips the sign
• Applying power rule when n = -1: forgetting that ∫x^(-1) dx = ln|x| + C, not x^0/0
• In substitution, forgetting to change the limits of integration in definite integrals — this leads to wrong answers
• Not simplifying the integrand before integrating — e.g., not splitting (3x^2 + 5x)/x into 3x + 5 first

• SIN-COS rule: SIN integrates to -COS (negative!), COS integrates to SIN (positive). Think: Sine gets a minus sign as a penalty.
• POWER RULE: x^n becomes x^(n+1)/(n+1). Raise the power by 1, divide by the new power. Exception: n = -1 becomes ln|x|.
• FTC (Fundamental Theorem): Find antiderivative F, plug in top limit, subtract bottom limit. Top minus Bottom = T - B.
• u-SUB: If you see a function inside another function, and the derivative of the inner one is nearby, use u-substitution. Look for the 'chain rule in reverse'.

• Agniveer Navy typically asks 2-3 integration questions per paper. Most are straightforward indefinite or definite integrals using basic formulas — no tricks.
• Trig integrals (sin, cos, sec^2) appear in ~40% of integration questions. Memorize the 6 standard trig integral formulas cold.
• Definite integrals with polynomial or exponential functions are very common. Practice evaluating F(b) - F(a) quickly and accurately.
• Substitution questions are rare (0-1 per paper) but when they appear, they are usually straightforward — the derivative of the inner function is explicitly present.
• Time management: integration questions should take 2-3 minutes each. If you're stuck, move on and return. Don't spend 5+ minutes on one integral.
• Watch for negative signs in trig integrals and when swapping limits — these are the most common careless errors in the exam hall.