∫xⁿ dx (n ≠ −1) equals:

xⁿ⁺¹/(n + 1) + c. Reverse of the power rule.

eˣ + c. eˣ is its own derivative.

aˣ/log a + c. Since d/dx aˣ = aˣ log a.

∫(3x² + 2x) dx equals:

x³ + x² + c. Integrate term by term.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 5: Integration

Short notes, key concepts, quick revision and practice MCQs with answers

Integration is the reverse process of differentiation. The chapter defines the indefinite integral, lists standard formulas and rules, and integrates by substitution, using special integrals, by parts and by partial fractions.

Key concepts: Integration

Indefinite integral: If d/dx g(x) = f(x), then ∫f(x) dx = g(x) + c, where c is the constant of integration. f(x) is the integrand.
Rules: ∫[f(x) ± g(x)] dx = ∫f dx ± ∫g dx; ∫k f(x) dx = k∫f(x) dx. If ∫f(x) dx = g(x) + c, then ∫f(ax + b) dx = (1/a) g(ax + b) + c.
Standard formulas: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c (n ≠ −1); ∫(1/x) dx = log|x| + c; ∫eˣ dx = eˣ + c; ∫aˣ dx = aˣ/log a + c; ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/(a(n + 1)) + c.
Method of substitution: Put a part of the integrand equal to t so that its derivative also appears. In particular ∫f′(x)/f(x) dx = log|f(x)| + c.
Special integrals: ∫dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + c; ∫dx/(a² − x²) = (1/2a) log|(a + x)/(a − x)| + c; ∫dx/√(x² + a²) = log|x + √(x² + a²)| + c; ∫dx/√(x² − a²) = log|x + √(x² − a²)| + c.
Integration by parts: ∫u v dx = u∫v dx − ∫[u′ ∫v dx] dx. Choose u as the function that becomes simpler on differentiation. Also ∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + c.
Partial fractions: A proper rational function is split into simpler fractions, e.g. 1/((x − 1)(x − 2)) = 1/(x − 2) − 1/(x − 1), and each part is integrated.

Quick revision: important points

Always add the constant of integration c.
∫log x dx = x log x − x + c.
∫x eˣ dx = eˣ(x − 1) + c.
∫1/x² dx = −1/x + c and ∫√x dx = (2/3)x^(3/2) + c.
∫e^(ax) dx = e^(ax)/a + c.

Practice MCQs with answers

∫xⁿ dx (n ≠ −1) equals:
1. n xⁿ⁻¹ + c
2. xⁿ⁺¹/(n + 1) + c
3. xⁿ⁺¹ + c
4. xⁿ/n + c
Show answer
Answer: B. xⁿ⁺¹/(n + 1) + c
Reverse of the power rule.
∫(1/x) dx equals:
1. −1/x² + c
2. log|x| + c
3. x + c
4. eˣ + c
Show answer
Answer: B. log|x| + c
Standard integral.
∫eˣ dx equals:
1. eˣ + c
2. x eˣ + c
3. eˣ⁺¹/(x + 1) + c
4. log x + c
Show answer
Answer: A. eˣ + c
eˣ is its own derivative.
∫aˣ dx equals:
1. aˣ log a + c
2. aˣ/log a + c
3. x aˣ⁻¹ + c
4. aˣ⁺¹ + c
Show answer
Answer: B. aˣ/log a + c
Since d/dx aˣ = aˣ log a.
∫(3x² + 2x) dx equals:
1. 6x + 2 + c
2. x³ + x² + c
3. 3x³ + 2x² + c
4. x³ + 2x² + c
Show answer
Answer: B. x³ + x² + c
Integrate term by term.

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