Is it correct to say: Forgetting the constant of integration +C in indefinite integrals?

Every indefinite integral MUST end in +C. It represents the family of antiderivatives.

Is it correct to say: Writing ∫ 1/x dx = ln x instead of ln|x|?

The absolute value is essential — the antiderivative must be defined for negative x as well.

Is it correct to say: Applying the power rule with n = −1?

The power rule requires n ≠ −1. For n = −1 use ∫ x^{-1} dx = ln|x| + C.

Is it correct to say: In integration by parts, choosing u as the harder function?

Use ILATE: pick u from the leftmost class present (Inverse, Logarithm, Algebraic, Trig, Exponential). This usually makes ∫v du easier than the original.

Is it correct to say: In substitution, changing x but forgetting to change dx to du?

u = g(x) ⇒ du = g'(x) dx. Substitute BOTH the function and its differential, or the limits (definite integral).

Is it correct to say: For definite integrals with substitution, forgetting to change limits?

Either change limits to u-values, OR back-substitute to x before evaluating limits. Do not mix the two.

Integrals

Indefinite and definite integrals, standard forms, substitution, integration by parts, partial fractions — NCERT Class 12 Maths Ch 7

📐 14 formulas✏️ 3 examples🎯 6 practice⚖️ 8-10 marks🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Integration by parts (ILATE) is asked every year — typically as a 3- or 5-mark question.
→Partial-fraction decomposition of a rational function shows up whenever the denominator factors nicely.
→For definite integrals, use the property that ∫₀^a f(x) dx = ∫₀^a f(a − x) dx to simplify symmetric integrands.
→Never forget the constant of integration +C in indefinite integrals — CBSE deducts a mark.
→For trig integrals, memorise ∫ sec²x dx = tan x + C and ∫ cosec²x dx = −cot x + C — small standard forms rescue big problems.

📐 Formulas(14)

Power Rule★ Board fav

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1

Integral of 1/x

\int \frac{1}{x} d x = ln ∣ x ∣ + C

Exponential and Logarithm

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

Standard Trig Integrals★ Board fav

\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

Inverse Trig Integrals★ Board fav

\int \frac{d x}{1 - x ^{2}} = sin^{- 1} x + C, \int \frac{d x}{1 + x ^{2}} = tan^{- 1} x + C

Integration by Substitution★ Board fav

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

Integration by Parts★ Board fav

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

Partial Fractions (Linear factors)★ Board fav

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

Standard Form: 1/(x² + a²)★ Board fav

\int \frac{d x}{x ^{2} + a ^{2}} = \frac{1}{a} tan^{- 1} (\frac{x}{a}) + C

Standard Form: 1/(x² − a²)

\int \frac{d x}{x ^{2} - a ^{2}} = \frac{1}{2 a} ln \frac{x - a}{x + a} + C

Standard Form: 1/√(a² − x²)

\int \frac{d x}{a ^{2} - x ^{2}} = sin^{- 1} (\frac{x}{a}) + C

Definite Integral (Fundamental Theorem)★ Board fav

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

Property: Reflection★ Board fav

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

Property: Even/Odd

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

✏️ Solved Examples

1Solved Exampleeasy3 steps

Evaluate ∫(3x² + 2x − 5) dx.

Apply the power rule term by term.

\int 3 x^{2} d x + \int 2 x d x - \int 5 d x

2Solved Exampleboard4 steps

Evaluate ∫ x · e^x dx.

ILATE: Algebraic (x) before Exponential (e^x), so u = x, dv = e^x dx.

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

3Solved ExampleHOTS4 steps

Evaluate ∫₀^{π/2} (sin x)/(sin x + cos x) dx.

Call the integral I and use the reflection property with a = π/2.

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

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Forgetting the constant of integration +C in indefinite integrals
✓Every indefinite integral MUST end in +C. It represents the family of antiderivatives.
2
Writing ∫ 1/x dx = ln x instead of ln|x|
✓The absolute value is essential — the antiderivative must be defined for negative x as well.
3
Applying the power rule with n = −1
✓The power rule requires n ≠ −1. For n = −1 use ∫ x^{-1} dx = ln|x| + C.
4
In integration by parts, choosing u as the harder function
✓Use ILATE: pick u from the leftmost class present (Inverse, Logarithm, Algebraic, Trig, Exponential). This usually makes ∫v du easier than the original.
5
In substitution, changing x but forgetting to change dx to du
✓u = g(x) ⇒ du = g'(x) dx. Substitute BOTH the function and its differential, or the limits (definite integral).
6
For definite integrals with substitution, forgetting to change limits
✓Either change limits to u-values, OR back-substitute to x before evaluating limits. Do not mix the two.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Evaluate ∫ (x³ + 4x − 1)/x² dx.
Q2
Evaluate ∫ x · sin x dx.
Q3
Evaluate ∫ (2x + 3)/(x² + 3x + 5) dx.
Q4
Evaluate ∫ dx/(x² + 6x + 13).
Q5
Evaluate ∫₀^1 x·e^{x²} dx.
Q6
Evaluate ∫ dx/((x − 1)(x − 2)) using partial fractions.

📝 Notes

Integrals

Integration reverses differentiation. Where the derivative measures instantaneous change, the integral accumulates it — giving area, distance, volume, work.

Two flavours of integral

Indefinite integral ∫f(x) dx = F(x) + C: family of antiderivatives.
Definite integral ∫_a^b f(x) dx = F(b) − F(a): a single number, the net signed area from a to b.

How to pick a technique

Polynomial or standard form — direct formula from the table.
You spot a function AND its derivative in the integrand — substitution.
Product of an algebraic and a transcendental function — integration by parts (ILATE).
Rational function P(x)/Q(x) with degree(P) < degree(Q) — partial fractions.
Quadratic denominator — complete the square, then reduce to arctan or log form.
Symmetric definite integral, limits 0 to a or −a to a — use the reflection or even/odd property.

The ILATE mnemonic

When choosing u in integration by parts, pick from the earliest available class:

I — inverse trig (sin⁻¹, tan⁻¹, ...)
L — logarithm (ln x, log x)
A — algebraic (x, x², polynomial)
T — trig (sin, cos, ...)
E — exponential (e^x, a^x)

Whichever appears earlier in ILATE becomes u; the other becomes dv.

Sanity checks

Differentiate your answer — you must get back the integrand.
Missing +C on an indefinite integral costs a mark every time.
If you replace x by (limit − x) in a definite integral and the integrand transforms nicely, the reflection property is likely the intended trick.