12th Mathematics (Arts & Science) Part 2 Chapter 3 Solution (Digest) Maharashtra state board

Chapter 3 Indefinite Integration

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Project on Indefinite Integration

Indefinite integration, also known as antiderivation or finding an antiderivative, is a fundamental concept in calculus. It involves determining a function whose derivative is the given function. The result of an indefinite integration is called an antiderivative or a primitive function. The process is essentially the reverse of differentiation.

Key Concepts

1. Definition:

   • The indefinite integral of a function $f(x)$ with respect to $x$ is denoted as $\int f(x) \, dx$.
   • It represents a family of functions $F(x) + C$, where $F'(x) = f(x)$ and $C$ is an arbitrary constant called the constant of integration.

2. Notation:

   • $\int f(x) \, dx = F(x) + C$
   • $\int$ is the integral sign, $f(x)$ is the integrand, $dx$ indicates the variable of integration, and $C$ is the constant of integration.

3. Basic Rules:

   • Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$
   • Constant Multiple Rule: $\int k f(x) \, dx = k \int f(x) \, dx$
   • Sum Rule: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$
   • Difference Rule: $\int [f(x) - g(x)] \, dx = \int f(x) \, dx - \int g(x) \, dx$

4. Common Integrals:

   • $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$
   • $\int e^x \, dx = e^x + C$
   • $\int \frac{1}{x} \, dx = \ln |x| + C$
   • $\int \sin x \, dx = -\cos x + C$
   • $\int \cos x \, dx = \sin x + C$
   • $\int \sec^2 x \, dx = \tan x + C$
   • $\int \csc^2 x \, dx = -\cot x + C$
   • $\int \sec x \tan x \, dx = \sec x + C$
   • $\int \csc x \cot x \, dx = -\csc x + C$

5. Techniques of Integration:

   • Substitution: Used when the integrand is a composite function. Set $u = g(x)$, then $du = g'(x) dx$.
     • Example: $\int 2x e^{x^2} \, dx$ set $u = x^2$, then $du = 2x \, dx$.
   • Integration by Parts: Based on the product rule for differentiation.
     • Formula: $\int u \, dv = uv - \int v \, du$
     • Example: $\int x e^x \, dx$, set $u = x$, $dv = e^x \, dx$.
   • Partial Fraction Decomposition: Used for rational functions, expressing them as a sum of simpler fractions.
     • Example: $\int \frac{1}{(x-1)(x+2)} \, dx$, decompose into $\int \left( \frac{A}{x-1} + \frac{B}{x+2} \right) \, dx$.
   • Trigonometric Integrals and Substitutions: Used for integrals involving trigonometric functions.
     • Example: $\int \sin^2 x \, dx$ using $\sin^2 x = \frac{1 - \cos(2x)}{2}$.
   • Improper Integrals: Dealing with integrals with infinite limits or integrands with infinite discontinuities. Not common in indefinite integration but relevant in the context of definite integrals.

Example Problem

Evaluate $\int (3x^2 + 2x + 1) \, dx$:

1. Apply the Sum Rule:

   $$\int (3x^2 + 2x + 1) \, dx = \int 3x^2 \, dx + \int 2x \, dx + \int 1 \, dx$$

2. Integrate Each Term:

   • $\int 3x^2 \, dx = 3 \cdot \frac{x^{3}}{3} = x^3$
   • $\int 2x \, dx = 2 \cdot \frac{x^{2}}{2} = x^2$
   • $\int 1 \, dx = x$

3. Combine Results and Add the Constant of Integration:

   $$x^3 + x^2 + x + C$$

Thus, $\int (3x^2 + 2x + 1) \, dx = x^3 + x^2 + x + C$.

Indefinite integration is a powerful tool in mathematics for solving problems involving rates of change and accumulation, and it is foundational for understanding and applying integral calculus.