Class 12 Maths Chapter 7: Integrals

Integrals is Chapter 7 of NCERT Class 12 Maths and one of the highest scoring chapters in the CBSE board exam. It treats integration as the reverse of differentiation, builds the main methods of finding indefinite integrals, and then uses the fundamental theorem of calculus to evaluate definite integrals with the help of their properties.

What you will learn

• Integration as anti-differentiation and the role of the constant of integration
• The standard integral formulas for powers, trigonometric, exponential and logarithmic functions
• Three key methods: substitution, partial fractions and integration by parts
• Integrals of special quadratic forms such as 1/(x² + a²) and 1/√(a² − x²)
• The fundamental theorem of calculus and the important properties of definite integrals

Key Concepts and Formulas

Indefinite Integral as an Antiderivative

If the derivative of F(x) is f(x), then F(x) is called an antiderivative of f(x). Since the derivative of a constant is zero, every function has infinitely many antiderivatives that differ only by a constant, so we always add C.

∫ f(x) dx = F(x) + C   (C = constant of integration)
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
∫ k f(x) dx = k ∫ f(x) dx

• Integration and differentiation are inverse processes of each other

Standard Integral Formulas

These results come straight from the derivatives you already know, so learn them as pairs.

∫ 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
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = −cosec x + C
∫ tan x dx = log|sec x| + C
∫ sec x dx = log|sec x + tan x| + C

Integration by Substitution

When the integrand contains a function and (a multiple of) its derivative, put the inner function equal to t. This turns a messy integral into a standard one, but the whole integral must be rewritten in t. Trigonometric identities often help before substituting.

If x = g(t), then dx = g′(t) dt
∫ f′(x)/f(x) dx = log|f(x)| + C
sin²x = (1 − cos 2x)/2

Integrals of Special Quadratic Forms

These six results are used again and again, especially after completing the square in ax² + bx + c.

∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
∫ dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + C
∫ dx/(a² − x²) = (1/2a) log|(a + x)/(a − x)| + C
∫ dx/√(a² − x²) = sin⁻¹(x/a) + C
∫ dx/√(x² + a²) = log|x + √(x² + a²)| + C
∫ dx/√(x² − a²) = log|x + √(x² − a²)| + C

Partial Fractions

A proper rational function P(x)/Q(x) whose denominator factorises can be split into simpler fractions, each of which integrates to a log or an inverse tan.

1/((x − a)(x − b)) = A/(x − a) + B/(x − b)
Repeated factor: A/(x − a) + B/(x − a)²

• If the degree of P(x) is not less than that of Q(x), divide first

Integration by Parts

Used for the product of two different kinds of functions. Choose the first function using the ILATE order: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.

∫ u v dx = u ∫ v dx − ∫ [u′ ∫ v dx] dx
∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C
∫ √(a² − x²) dx
  = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C

Definite Integrals and Their Properties

A definite integral gives a number, not a family of functions. The fundamental theorem of calculus links it to antiderivatives, where F is any antiderivative of f.

∫ₐᵇ f(x) dx = F(b) − F(a)
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
Even f: ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx
Odd f:  ∫₋ₐᵃ f(x) dx = 0

• For any point c between a and b, the integral from a to b splits into the integral from a to c plus the integral from c to b, which is useful for modulus and piecewise functions

Solved Examples

Example 1: Evaluate ∫ (6x² − 4x + 5) dx.

= 6(x³/3) − 4(x²/2) + 5x + C
∴ ∫ (6x² − 4x + 5) dx = 2x³ − 2x² + 5x + C

Example 2: Evaluate ∫ 2x cos(x² + 3) dx.

Put x² + 3 = t, so 2x dx = dt
I = ∫ cos t dt = sin t + C
∴ I = sin(x² + 3) + C

Example 3: Evaluate ∫ dx/(x² + 6x + 13).

x² + 6x + 13 = (x + 3)² + 2²
I = ∫ dx/((x + 3)² + 2²)
∴ I = (1/2) tan⁻¹((x + 3)/2) + C

Example 4: Evaluate ∫ dx/((x − 1)(x + 3)).

1/((x − 1)(x + 3)) = A/(x − 1) + B/(x + 3)
1 = A(x + 3) + B(x − 1)
x = 1:  1 = 4A, so A = 1/4
x = −3: 1 = −4B, so B = −1/4
I = (1/4) log|x − 1| − (1/4) log|x + 3| + C
∴ I = (1/4) log|(x − 1)/(x + 3)| + C

Example 5: Evaluate ∫ x e²ˣ dx.

u = x (algebraic), v = e²ˣ
I = x(e²ˣ/2) − ∫ 1 × (e²ˣ/2) dx
∫ (e²ˣ/2) dx = (1/4)e²ˣ
∴ I = (x/2)e²ˣ − (1/4)e²ˣ + C

Example 6: Evaluate ∫₀² (3x² + 2x) dx.

Antiderivative = x³ + x²
= (2³ + 2²) − (0 + 0)
= 8 + 4
∴ ∫₀² (3x² + 2x) dx = 12

Example 7: Evaluate I = ∫₀^(π/2) sin⁷x/(sin⁷x + cos⁷x) dx.

Use ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
I = ∫₀^(π/2) cos⁷x/(cos⁷x + sin⁷x) dx
Adding: 2I = ∫₀^(π/2) 1 dx = π/2
∴ I = π/4

Common Mistakes to Avoid

• Forgetting + C in indefinite integrals costs marks; never add C to a definite integral.
• After substitution in a definite integral, either change the limits to the new variable or convert back to x before substituting limits, but do not mix the two.
• In integration by parts, choosing the wrong first function makes the integral harder; follow ILATE unless the question clearly suggests otherwise.
• Writing ∫ (1/x) dx as log x instead of log|x| is marked as incomplete.

Quick Revision

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C,  n ≠ −1
∫ (1/x) dx = log|x| + C
∫ uv dx = u∫v dx − ∫[u′∫v dx] dx
∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
∫ₐᵇ f(x) dx = F(b) − F(a)
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx

• Substitution: look for a function and its derivative together
• Parts: choose the first function using ILATE

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