Rehaan — Master Integrals

Chapter 01

What Are Integrals?

An integral is one of the two fundamental operations in calculus (the other being the derivative). While derivatives measure the rate of change of a function, integrals measure the accumulation of quantities. Think of it this way: if a derivative breaks things apart, an integral puts them back together.

The most intuitive way to understand an integral is as the area under a curve. Given a function \( f(x) \) on an interval \([a, b]\), the integral computes the total area between the curve and the x-axis.

The Integral Symbol $$\int f(x)\, dx$$

The elongated "S" symbol (\(\int\)) stands for summation — it was introduced by Leibniz as a stylized "S" for "summa." The \(dx\) indicates we are integrating with respect to the variable \(x\), and \(f(x)\) is the function being integrated, called the integrand.

Chapter 02

Indefinite Integrals

An indefinite integral represents a family of functions whose derivative equals the integrand. It is written without bounds and always includes a constant of integration \(C\), because the derivative of any constant is zero.

General Form $$\int f(x)\, dx = F(x) + C$$

Here, \(F(x)\) is called an antiderivative of \(f(x)\). This means that \(F'(x) = f(x)\). The constant \(C\) accounts for the fact that there are infinitely many antiderivatives that differ only by a constant.

Example — Power Rule

Problem: Find \(\displaystyle\int x^3\, dx\)

Power Rule: \(\displaystyle\int x^n\, dx = \frac{x^{n+1}}{n+1} + C\) (where \(n \neq -1\))

Apply: \(\displaystyle\int x^3\, dx = \frac{x^{4}}{4} + C\)

Verify: \(\displaystyle\frac{d}{dx}\left(\frac{x^4}{4} + C\right) = x^3\) ✓

Example — Sum Rule

Problem: Find \(\displaystyle\int (3x^2 + 2x - 5)\, dx\)

Split: \(\displaystyle 3\int x^2\, dx + 2\int x\, dx - 5\int 1\, dx\)

Evaluate: \(\displaystyle 3 \cdot \frac{x^3}{3} + 2 \cdot \frac{x^2}{2} - 5x + C = x^3 + x^2 - 5x + C\)

Chapter 03

Definite Integrals

A definite integral computes the net area between the curve \(f(x)\) and the x-axis over a specific interval \([a, b]\). Unlike indefinite integrals, the result is a number, not a function.

The Fundamental Theorem of Calculus $$\int_a^b f(x)\, dx = F(b) - F(a)$$

This is one of the most important results in all of mathematics. It connects the concept of an antiderivative with the computation of area. To evaluate a definite integral, find the antiderivative \(F(x)\), plug in the upper bound \(b\), plug in the lower bound \(a\), and subtract.

Example — Evaluating a Definite Integral

Problem: Evaluate \(\displaystyle\int_1^3 2x\, dx\)

Antiderivative: \(F(x) = x^2\)

Evaluate: \(F(3) - F(1) = 9 - 1 = 8\)

Interpretation: The area under \(2x\) from \(x=1\) to \(x=3\) is 8 square units.

Important note: Definite integrals measure net area. If \(f(x)\) dips below the x-axis, that region counts as negative area. To find total area regardless of sign, integrate \(|f(x)|\) instead.

Chapter 04

U-Substitution

U-substitution is the integration counterpart of the chain rule for derivatives. When an integrand is a composition of functions, we can simplify it by substituting a new variable \(u\) for part of the expression.

The Technique $$\int f(g(x)) \cdot g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)$$

Example — Basic U-Substitution

Problem: Find \(\displaystyle\int 2x \cdot e^{x^2}\, dx\)

Choose u: Let \(u = x^2\), so \(du = 2x\, dx\)

Substitute: \(\displaystyle\int e^{u}\, du\)

Integrate: \(e^u + C\)

Back-substitute: \(e^{x^2} + C\)

Example — Adjusting Constants

Problem: Find \(\displaystyle\int x \cos(x^2)\, dx\)

Choose u: Let \(u = x^2\), so \(du = 2x\, dx\), meaning \(x\, dx = \frac{1}{2}\, du\)

Substitute: \(\displaystyle\frac{1}{2}\int \cos(u)\, du\)

Integrate: \(\displaystyle\frac{1}{2}\sin(u) + C\)

Back-substitute: \(\displaystyle\frac{1}{2}\sin(x^2) + C\)

Reference

Key Integral Formulas

Keep these fundamental integrals at your fingertips. Every complex integral ultimately reduces to combinations of these building blocks.

Function	Integral
\(\displaystyle x^n\)	\(\displaystyle \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)\)
\(\displaystyle \frac{1}{x}\)	\(\displaystyle \ln\|x\| + C\)
\(\displaystyle e^x\)	\(\displaystyle e^x + C\)
\(\displaystyle a^x\)	\(\displaystyle \frac{a^x}{\ln a} + C\)
\(\displaystyle \sin x\)	\(\displaystyle -\cos x + C\)
\(\displaystyle \cos x\)	\(\displaystyle \sin x + C\)
\(\displaystyle \sec^2 x\)	\(\displaystyle \tan x + C\)
\(\displaystyle \frac{1}{1+x^2}\)	\(\displaystyle \arctan x + C\)
\(\displaystyle \frac{1}{\sqrt{1-x^2}}\)	\(\displaystyle \arcsin x + C\)

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