Class 12 Maths Integrals Notes

7.1 Introduction

Integration is the inverse process of differentiation and is used to calculate areas, volumes, and solve real-life problems involving accumulation.

• If differentiation measures rate of change, integration measures total accumulation.
• Integration is widely applied in physics, engineering, and economics.

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7.2 Integration as an Inverse Process of Differentiation

• If $F(x)$F(x) is a function such that $F'(x) = f(x)$F′(x)=f(x), then F(x) is an antiderivative of f(x).
• The most general form of antiderivative:

$$\int f(x) dx = F(x) + C$$∫f(x)dx=F(x)+C

where C is the constant of integration.

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7.3 Methods of Integration

🔹 1. Substitution Method

• Replace part of the integrand with a single variable to simplify integration.

🔹 2. Integration by Parts

• Based on the formula:

$$\int u\,dv = uv – \int v\,du$$∫udv=uv−∫vdu

🔹 3. Partial Fractions

• Break a rational function into simpler fractions to integrate easily.

🔹 4. Trigonometric and Special Methods

• Use identities and standard forms to solve integrals.

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7.4 Integrals of Some Particular Functions

Common formulas include:$$\int x^n dx = \frac{x^{n+1}}{n+1}, \quad n \neq -1$$∫xndx=n+1xn+1​,n=−1 $$\int e^x dx = e^x, \quad \int \frac{1}{x} dx = \ln|x|$$∫exdx=ex,∫x1​dx=ln∣x∣ $$\int \sin x dx = -\cos x, \quad \int \cos x dx = \sin x$$∫sinxdx=−cosx,∫cosxdx=sinx

These are essential for solving many problems.

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7.5 Integration by Partial Fractions

• Express a rational function $P(x)/Q(x)$P(x)/Q(x) as the sum of simpler fractions.
• Then integrate each fraction individually.

Example:$$\frac{3x+5}{x^2 – x – 2} = \frac{A}{x-2} + \frac{B}{x+1}$$x2−x−23x+5​=x−2A​+x+1B​

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7.6 Integration by Parts

• Useful for products of functions (e.g., $x e^x$xex, $x \ln x$xlnx).
• Formula:

$$\int u\, dv = uv – \int v\, du$$∫udv=uv−∫vdu

• Choose u as the part that becomes simpler after differentiation.

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7.7 Definite Integral

A definite integral has upper and lower limits:$$\int_a^b f(x) dx$$∫ab​f(x)dx

• Represents the net area under the curve from x = a to x = b.

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7.8 Fundamental Theorem of Calculus

1. If $F(x)$F(x) is an antiderivative of f(x):

$$\int_a^b f(x) dx = F(b) – F(a)$$∫ab​f(x)dx=F(b)−F(a)

2. Connects differentiation and integration.

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7.9 Evaluation of Definite Integrals by Substitution

• Substitute $x = g(t)$x=g(t) to simplify the integral.
• Also known as change of variable method.

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7.10 Some Properties of Definite Integrals

• Linearity: $\int_a^b [cf(x) + g(x)] dx = c \int_a^b f(x) dx + \int_a^b g(x) dx$∫ab​[cf(x)+g(x)]dx=c∫ab​f(x)dx+∫ab​g(x)dx
• Additivity: $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$∫ab​f(x)dx=∫ac​f(x)dx+∫cb​f(x)dx
• Reversal of limits: $\int_a^b f(x) dx = -\int_b^a f(x) dx$∫ab​f(x)dx=−∫ba​f(x)dx

These properties simplify complex integral calculations.

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✅ Key Points to Remember

• Integration is the inverse of differentiation.
• Indefinite integral always includes constant C.
• Definite integrals give area under the curve.
• Methods include substitution, parts, partial fractions.
• Fundamental theorem connects derivatives and integrals.