Basic of Concept Anti-derivative

MATHEMATICS GRADE 11 / 12

Antiderivative

Chapter: Integration

BY: GHANASHYAM ADHIKARI

GA

SIZE: 1 px 2 px 3 px 4 px

History of Antiderivatives & Integration

The history of integration dates back to ancient Greece with Archimedes using the method of exhaustion to calculate areas. However, the modern concept of the antiderivative was established independently in the late 17th century by Sir Isaac Newton and Gottfried Wilhelm Leibniz.

Gottfried Wilhelm Leibniz (1646–1716)

Introduced the integral symbol $\int$ representing an elongated 'S' for summa (sum), as well as the differential notation $dx$.

Bernhard Riemann (1826–1866)

Formulated the rigorous definition of the definite integral as the limit of Riemann Sums of rectangles under a function curve.

Real-Life Example: Hydraulic Integrator (Flow vs. Volume)

Integration measures accumulation! The flow rate of water $f(t)$ into a tank is the derivative of the water volume $V(t)$. Thus, Volume is the integral of flow rate plus initial constant $+ c$.

Flow Rate $f(t) = 2x$ (Faucet Opening):

Initial Water Constant $+ c$ (In Bucket):

Note: Suppose the can already has water in it, let it be $c$ value, which gives us to always add $+ c$.

Integration as Riemann Sum

Partition of interval $[a, b]$ into $N$ sub-rectangles: $x_0 = a < x_1 < \dots < x_N = b$

Subdivisions $N$: 12

$$\int_a^b f(x)dx \approx \sum_{i=1}^N f(x_i^*)(x_i - x_{i-1})$$

EXACT INTEGRAL AREA: 18.600

RIEMANN SUM APPROXIMATION: 18.820

Definition of Antiderivative & Indefinite Integral

Definition:

A function $F(x)$ is called an antiderivative or primitive of a function $f(x)$ on an interval $I$ if:

$$F'(x) = \frac{d}{dx}[F(x)] = f(x) \quad \text{for all } x \in I$$

Indefinite Integral Notation:

The process of finding antiderivatives is called Integration. It is denoted as:

$$\int f(x) \, dx = F(x) + c$$

• $\int$ = Integral symbol (elongated S for summa)
• $f(x)$ = Integrand
• $dx$ = Variable of integration
• $F(x)$ = Anti-derivative / Primitive
• $c$ = Arbitrary constant of integration

Derivation of Fundamental Integration Formulas

1. Derivation of the Power Rule:

From differential calculus, we know that for any real power $n \neq -1$:

$$\frac{d}{dx} \left[ \frac{x^{n+1}}{n+1} \right] = \frac{1}{n+1} \cdot (n+1) x^{(n+1)-1} = x^n$$

By definition of Antiderivative, integrating both sides:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + c \quad (n \neq -1)$$

2. Derivation for Trigonometric Antiderivatives:

Since $\frac{d}{dx}(\sin x) = \cos x \implies \int \cos x \, dx = \sin x + c$

Since $\frac{d}{dx}(-\cos x) = \sin x \implies \int \sin x \, dx = -\cos x + c$

Since $\frac{d}{dx}(\tan x) = \sec^2 x \implies \int \sec^2 x \, dx = \tan x + c$

Integrations Standard Formulas

Algebraic & Exponential Rules

$\int 1 \, dx = x + c$

$\int a \, dx = ax + c$

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + c, \quad n \neq -1$

$\int \frac{1}{x} \, dx = \ln|x| + c, \quad x \neq 0$

$\int e^x \, dx = e^x + c$

$\int e^{ax+b} \, dx = \frac{e^{ax+b}}{a} + c$

$\int a^x \, dx = \frac{a^x}{\ln a} + c, \quad a > 0, a \neq 1$

$\int (ax+b)^n \, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c$

$\int \frac{1}{ax+b} \, dx = \frac{\ln|ax+b|}{a} + c$

Trigonometric Rules

$\int \cos x \, dx = \sin x + c$

$\int \sin x \, dx = -\cos x + c$

$\int \sec^2 x \, dx = \tan x + c$

$\int \csc^2 x \, dx = -\cot x + c$

$\int \sec x \tan x \, dx = \sec x + c$

$\int \csc x \cot x \, dx = -\csc x + c$

Class 11 Integration Examination Quiz

Select the correct antiderivative for each problem from the exercise sheet.

Score: 0 / 10

Worked Out Examples (Exercise 1)

1. Integrate: $$\int 5x^4 \, dx$$

Solution: $\int 5x^4 \, dx$

$= 5 \int x^4 \, dx$

$= 5 \times \frac{x^{4+1}}{4+1} + c$

$= \frac{5x^5}{5} + c = x^5 + c$

2. Evaluate: $$\int (3x + 4)^{-2/3} \, dx$$

Solution: $\int (3x+4)^{-2/3} \, dx$

$= \frac{(3x+4)^{-(2/3)+1}}{3 \left( -\frac{2}{3} + 1 \right)} + c$

$= \frac{(3x+4)^{1/3}}{3 \left( \frac{1}{3} \right)} + c = (3x+4)^{1/3} + c$

3. Evaluate: $$\int \frac{x^3 - 1}{x - 1} \, dx$$

Solution: $\int \frac{x^3-1}{x-1} \, dx$

$= \int \frac{(x-1)(x^2 + x + 1)}{(x-1)} \, dx \quad [\because a^3 - b^3 = (a-b)(a^2+ab+b^2)]$

$= \int (x^2 + x + 1) \, dx$

$= \int x^2 \, dx + \int x \, dx + \int 1 \, dx$

$= \frac{x^3}{3} + c_1 + \frac{x^2}{2} + c_2 + x + c_3$

$= \frac{x^3}{3} + \frac{x^2}{2} + x + c$, where $c = c_1 + c_2 + c_3$

Note: Generally, we write only one constant $c$ instead of writing all constants after integrating of indefinite integral.

Solution by Ghanashyam Adhikari

Practice Problems (Exercise 2)

1. Evaluate: $$\int (\sin x + \cos x) \, dx$$

Solution:

$= \int \sin x \, dx + \int \cos x \, dx$

$= -\cos x + \sin x + c = \sin x - \cos x + c$

2. Evaluate: $$\int e^{2x+3} \, dx$$

Solution:

Using formula $\int e^{ax+b} dx = \frac{e^{ax+b}}{a} + c$:

$= \frac{e^{2x+3}}{2} + c$

3. Evaluate: $$\int \frac{1}{5x - 2} \, dx$$

Solution:

Using formula $\int \frac{1}{ax+b} dx = \frac{\ln|ax+b|}{a} + c$:

$= \frac{\ln|5x - 2|}{5} + c$

Solution by Ghanashyam Adhikari