Basics of Integration · Ch.7

01

What is Integration?

Core Concept

Integration is the reverse process of differentiation. If the derivative of \(F(x)\) is \(f(x)\), then the integral of \(f(x)\) is \(F(x)\) plus a constant \(C\).

FUNDAMENTAL RELATIONSHIP

\[ \frac{d}{dx}\big[F(x)\big] = f(x) \;\implies\; \int f(x)\,dx = F(x) + C \]

Here, \(F(x)\) is called the antiderivative (or primitive) of \(f(x)\), and \(C\) is the constant of integration.

Terminology

Symbol / Term	Meaning
\(\int\)	Integral sign (elongated S, from Latin summa)
\(f(x)\)	Integrand — the function being integrated
\(dx\)	Indicates integration with respect to \(x\)
\(C\)	Constant of Integration — added because derivative of constant is 0
\(F(x)+C\)	General Integral (indefinite integral)

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Why +C? Since the derivative of any constant is zero, there are infinitely many antiderivatives differing only by a constant. For example, \(\int 2x\,dx = x^2 + 5\) or \(x^2 - 3\) or \(x^2 + 100\) — all are valid. We write \(x^2 + C\) to represent all of them.

02

Geometrical Interpretation

Area Under Curve

Geometrically, the definite integral \(\int_a^b f(x)\,dx\) represents the net area bounded by the curve \(y = f(x)\), the \(x\)-axis, and the vertical lines \(x = a\) and \(x = b\).

AREA INTERPRETATION

\[ \text{Area} = \int_a^b f(x)\,dx \quad \text{(when } f(x) \geq 0 \text{ on } [a,b]) \]

The indefinite integral \(\int f(x)\,dx = F(x)+C\) gives a family of curves, each shifted vertically by the constant \(C\).

03

Rules of Integration

1

Constant Multiple Rule

A constant factor can be taken out of the integral sign.

\[ \int k\cdot f(x)\,dx = k\int f(x)\,dx, \quad k \in \mathbb{R} \]
2

Sum / Difference Rule

The integral of a sum (or difference) equals the sum (or difference) of the individual integrals.

\[ \int \big[f(x) \pm g(x)\big]\,dx = \int f(x)\,dx \;\pm\; \int g(x)\,dx \]
3

Linearity (Combined)

Both rules together make integration a linear operation.

\[ \int \big[af(x) + bg(x)\big]\,dx = a\int f(x)\,dx + b\int g(x)\,dx \]

04

Elementary Standard Formulas

Power & Algebraic

POWER RULE (n ≠ −1)

\(\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\)

RECIPROCAL

\(\displaystyle\int \frac{1}{x}\,dx = \ln|x|+C\)

CONSTANT

\(\displaystyle\int k\,dx = kx+C\)

Exponential & Logarithmic

e^x

\(\displaystyle\int e^x\,dx = e^x+C\)

a^x

\(\displaystyle\int a^x\,dx = \frac{a^x}{\ln a}+C\)

Trigonometric Basics

SIN x

\(\displaystyle\int \sin x\,dx = -\cos x+C\)

COS x

\(\displaystyle\int \cos x\,dx = \sin x+C\)

SEC² x

\(\displaystyle\int \sec^2 x\,dx = \tan x+C\)

COSEC² x

\(\displaystyle\int \csc^2 x\,dx = -\cot x+C\)

SEC x · TAN x

\(\displaystyle\int \sec x\tan x\,dx = \sec x+C\)

COSEC x · COT x

\(\displaystyle\int \csc x\cot x\,dx = -\csc x+C\)

Inverse Trigonometric

1/√(1−x²)

\(\displaystyle\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x+C\)

−1/√(1−x²)

\(\displaystyle\int \frac{-1}{\sqrt{1-x^2}}\,dx = \cos^{-1}x+C\)

1/(1+x²)

\(\displaystyle\int \frac{1}{1+x^2}\,dx = \tan^{-1}x+C\)

1/(x√(x²−1))

\(\displaystyle\int \frac{1}{x\sqrt{x^2-1}}\,dx = \sec^{-1}x+C\)

05

Worked Examples

Example 1 — Power Rule

Find \(\displaystyle\int x^5\,dx\)

\[ \int x^5\,dx = \frac{x^{5+1}}{5+1}+C = \frac{x^6}{6}+C \]

Example 2 — Sum Rule

Find \(\displaystyle\int (3x^2 - 4\cos x + e^x)\,dx\)

\[ = 3\cdot\frac{x^3}{3} - 4\sin x + e^x + C = x^3 - 4\sin x + e^x + C \]

Example 3 — Rewrite First

Find \(\displaystyle\int \sqrt{x}\,dx\)

Rewrite \(\sqrt{x} = x^{1/2}\), then apply the power rule:

\[ \int x^{1/2}\,dx = \frac{x^{3/2}}{3/2}+C = \frac{2}{3}x^{3/2}+C \]

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Key Tip: Always simplify or rewrite the integrand into standard forms before integrating — e.g., \(\dfrac{1}{x^3} = x^{-3}\), \(\sqrt[3]{x^2} = x^{2/3}\).