Is it correct to say: Forgetting +C in indefinite integrals?

Always add +C. HSC marking scheme deducts a mark otherwise.

Is it correct to say: Applying substitution but not changing limits (definite integrals)?

When u = g(x), new limits are g(a) and g(b). Do NOT leave old limits.

Is it correct to say: Choosing u incorrectly in integration by parts?

Follow ILATE: pick u from higher in the list (Inverse trig, Log, Algebraic, Trig, Exp).

Is it correct to say: Applying partial fractions to improper rational function?

If deg(P) ≥ deg(Q), first divide P/Q and apply partial fractions on the remainder.

Is it correct to say: Forgetting minus sign in integral of tan x or 1/cos²x?

∫tan x dx = −ln|cos x| + C. ∫sec²x dx = tan x + C. Memorise signs.

Is it correct to say: Using ∫sin x cos x dx as substitution but writing u = sin x du = sin x dx?

u = sin x ⇒ du = cos x dx, not sin x dx. Check du carefully.

Integration

Indefinite integration, substitution, partial fractions, by parts, definite integrals — Maharashtra HSC Maths Ch 5 & 6

📐 12 formulas✏️ 3 examples🎯 5 practice⚖️ 10 marks🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→Integration by parts (∫u dv = uv − ∫v du) is a Maharashtra HSC 4-marker; remember ILATE rule to choose u.
→Definite integral properties (especially ∫[0 to a] f(a−x) dx and ∫[−a to a] even/odd) are asked as 3-marker every year.
→Partial fractions method for ∫dx/((x−a)(x−b)) is a routine 3-marker; check discriminant type before decomposing.
→Substitution: trigonometric x = a sin θ for √(a²−x²); x = a tan θ for √(a²+x²); x = a sec θ for √(x²−a²).
→Definite integral as limit of a sum (Riemann sum) is asked once or twice a year for 4 marks; know the formula for Σr, Σr², Σr³.

📐 Formulas(12)

Fundamental Theorem of Calculus★ Board fav

\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} (x) = f (x)

Standard Integrals★ Board fav

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1); \int \frac{d x}{x} = ln ∣ x ∣ + C

Trigonometric Integrals

\int sin x d x = - cos x + C; \int cos x d x = sin x + C; \int sec^{2} x d x = tan x + C

Exponential and Log Integrals

\int e^{x} d x = e^{x} + C; \int a^{x} d x = \frac{a ^{x}}{l n a} + C

Integration by Substitution★ Board fav

\int f (g (x)) g^{'} (x) d x = \int f (u) d u with u = g (x)

Integration by Parts (ILATE)★ Board fav

\int u d v = uv - \int v d u

Partial Fractions★ Board fav

\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

Definite Integral Property (1)★ Board fav

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

Definite Integral Property (2 — even/odd)

\int_{- a}^{a} f (x) d x = {2 \int_{0}^{a} f (x) d x 0 f even f odd

Reduction Formulae

\int sin^{n} x d x = - \frac{s i n ^{n - 1} x c o s x}{n} + \frac{n - 1}{n} \int sin^{n - 2} x d x

Definite Integral as Limit of a Sum★ Board fav

\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{r = 1}^{n} f (a + r h) \cdot h, h = \frac{b - a}{n}

Common Trigonometric Substitutions

a^{2} - x^{2} : x = a sin θ; a^{2} + x^{2} : x = a tan θ; x^{2} - a^{2} : x = a sec θ

✏️ Solved Examples

1Solved Exampleeasy3 steps

Evaluate ∫x eˣ dx.

ILATE: choose u = x (algebraic), dv = eˣ dx.

u = x, d v = e^{x} d x; d u = d x, v = e^{x}

2Solved Exampleboard3 steps

Evaluate ∫[0 to π/2] sin²x dx.

Use identity sin²x = (1 − cos 2x)/2.

\int_{0}^{π /2} sin^{2} x d x = \int_{0}^{π /2} \frac{1 - c o s 2 x}{2} d x

3Solved ExampleHOTS5 steps

Prove that ∫[0 to π] x sin x/(1 + cos²x) dx = π²/4.

Let I = ∫₀^π x sin x/(1 + cos²x) dx. Use property ∫₀^a f(x)dx = ∫₀^a f(a−x)dx.

I = \int_{0}^{π} \frac{( π - x ) s i n ( π - x )}{1 + c o s ^{2} ( π - x )} d x = \int_{0}^{π} \frac{( π - x ) s i n x}{1 + c o s ^{2} x} d x

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Forgetting +C in indefinite integrals
✓Always add +C. HSC marking scheme deducts a mark otherwise.
2
Applying substitution but not changing limits (definite integrals)
✓When u = g(x), new limits are g(a) and g(b). Do NOT leave old limits.
3
Choosing u incorrectly in integration by parts
✓Follow ILATE: pick u from higher in the list (Inverse trig, Log, Algebraic, Trig, Exp).
4
Applying partial fractions to improper rational function
✓If deg(P) ≥ deg(Q), first divide P/Q and apply partial fractions on the remainder.
5
Forgetting minus sign in integral of tan x or 1/cos²x
✓∫tan x dx = −ln|cos x| + C. ∫sec²x dx = tan x + C. Memorise signs.
6
Using ∫sin x cos x dx as substitution but writing u = sin x du = sin x dx
✓u = sin x ⇒ du = cos x dx, not sin x dx. Check du carefully.

🎯 Practice Yourself

🎯Practice Yourself5 questions

Q1
Evaluate ∫x² eˣ dx.
Q2
Evaluate ∫dx/(x² − 4).
Q3
Evaluate ∫[0 to π] sin x/(1 + cos²x) dx.
Q4
Evaluate ∫sec²x/(1 + tan x) dx.
Q5
Evaluate ∫[0 to 1] x/(1 + x²) dx.

📝 Notes

Integration — Maharashtra HSC Overview

Chapters 5 and 6 of the Balbharati Class 12 Mathematics textbook together cover indefinite integration, definite integrals, methods of integration, and properties.

Maharashtra syllabus specifics

Definite integral as limit of a sum (Riemann definition) is a Maharashtra HSC 4-mark question — write the Σ carefully and use Σr, Σr² formulas.
Special property questions (e.g. ∫₀^a f(x) = ∫₀^a f(a−x)) are asked every year — usually as HOTS.
Partial fractions with all four cases (distinct linear, repeated linear, irreducible quadratic, repeated quadratic) is explicitly asked.
Integration by parts with logarithmic integrand (e.g. ∫log x dx, ∫x log x dx) is a Balbharati staple.
The Maharashtra textbook lists a table of standard formulas — many with 1-mark direct application questions.

Strategy

ILATE priority for integration by parts.
For √(a² − x²), substitute x = a sin θ. For √(a² + x²), x = a tan θ. For √(x² − a²), x = a sec θ.
Change limits when substituting in definite integrals.
Always add +C for indefinite integrals.
For definite integrals, use properties to simplify BEFORE integrating.