Integration is a fundamental concept in calculus, often seen as the inverse operation of differentiation. There are several key rules and techniques for integration, each useful in different situations. Here’s an overview of the most important ones:

## Basic Integration Rules

- Constant Rule: $\int a \, dx = ax + C$, where $a$ is a constant and $C$ is the constant of integration.
- Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
- Sum Rule: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$.
- Difference Rule: $\int [f(x) – g(x)] \, dx = \int f(x) \, dx – \int g(x) \, dx$.
- Constant factor rule : $ \int a f(x) \; dx =a \int f(x) \; dx$.
- Generalized rule: $ \int a_1 f_1(x) + a_2 f_2(x) + a_3 f_3(x) \; dx =a_1 \int f_1(x) \; dx +a_2 \int f_2(x) \; dx + a_3 \int f_3(x) \; dx $

## Integration of Trigonometric Functions

- $\int \sin x \, dx = -\cos x + C$.
- $\int \cos x \, dx = \sin x + C$.
- $\int \sec^2 x \, dx = \tan x + C$.
- $\int \csc^2 x \, dx = -\cot x + C$.
- $\int \sec x \tan x \, dx = \sec x + C$.
- $\int \csc x \cot x \, dx = -\csc x + C$

## Exponential and Logarithmic Functions

- $\int e^x \, dx = e^x + C$.
- $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$ for $a > 0, a \neq 1$.
- $\int \ln x \, dx = x \ln x – x + C$ for $x > 0$.

## Integration by Substitution

Used when an integral contains a function and its derivative. It’s similar to the chain rule in differentiation

- $\int f(g(x))g'(x) \, dx = \int f(u) \, du$ where $u = g(x)$.

**Example**

$$

\int 2x \sqrt{x^2 + 1} \, dx

$$

Set ( $u = x^2 + 1$ ), then ( $du = 2x \, dx$ ). The integral becomes:

$$

\int \sqrt{u} \, du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(x^2 + 1)^{3/2} + C

$$

## Integration by Parts

Based on the product rule for differentiation, used for integrals of products of functions:

- $\int u \, dv = uv – \int v \, du$, where $u$ and $v$ are functions of $x$.

**Example**

$$

\int x e^x \, dx

$$

Choose ( u = x ) and ( $dv = e^x \, dx$ ). Then ( du = dx ) and ( $v = e^x$ ). The formula gives:

$$

x e^x – \int e^x \, dx = x e^x – e^x + C

$$

## Trigonometric Substitution

Used for integrals involving square roots of quadratic expressions, particularly useful for expressions like $\sqrt{a^2 – x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 – a^2}$.

## Partial Fractions

Used for integrating rational functions (fractions where the numerator and denominator are polynomials). The fraction is expressed as a sum of simpler fractions which can be integrated individually.

$ \int \frac {px +q}{(x-a)(x-b)} dx =\int \left \{ \frac {A}{x-a} + \frac {B}{x-b} \right \} dx$

$ \int \frac {px^2 +qx + r}{(x-a)(x-b)(x-c)} dx =\int \left \{ \frac {A}{x-a} + \frac {B}{x-b} + \frac {C}{x-c} \right \} dx$

$ \int \frac {px +q}{(x-a)^2} dx =\int \left \{ \frac {A}{x-a} + \frac {B}{(x-a)^2} \right \} dx$

$ \int \frac {px^2 +qx + r}{(x-a)^2(x-c)} dx =\int \left \{ \frac {A}{x-a} + \frac {B}{(x-a)^2} + \frac {C}{x-c} \right \} dx$

$ \int \frac {px^2 +q+r}{(x-a)(x^2 + bx +c)} dx =\int \left \{ \frac {A}{x-a} + \frac {Bx +C}{x^2 + bx +c} \right \} dx$

where $x^2 + bx +c$ is a irreducible quadratic

## Definite Integrals

- $\int_{a}^{b} f(x) \, dx$, where $a$ and $b$ are the limits of integration.

Each of these Integration rules and techniques can be applied in different scenarios, depending on the nature of the function to be integrated. Understanding when and how to use each rule is a key part of mastering integral calculus.

## FAQ on Integration

**Q 1:** What is the difference between definite and indefinite integrals?

**Answer:** An indefinite integral, represented as $\int f(x) \, dx$, is the general form of the antiderivative of $f(x)$ and includes a constant of integration $C$. A definite integral, represented as $\int_{a}^{b} f(x) \, dx$, computes the net area under the curve of $f(x)$ from $x = a$ to $x = b$ and does not include the constant of integration.

**Q 2:**How do you choose the method for integration?

**Answer: **The choice of integration method depends on the form of the integrand. For basic functions, apply standard formulas. If the integrand is a product of functions, integration by parts may be suitable. For a composite function, consider substitution. For rational functions, partial fractions can be effective. Trigonometric substitution is helpful for integrals involving square roots of quadratic expressions.

**Q3:**Can every function be integrated?

**Answer**: Not all functions have elementary antiderivatives. Some functions require special functions for their representation or numerical methods for their evaluation.

**Q4: **What is the significance of the constant of integration?

**Answer**: The constant of integration in indefinite integrals represents the family of all antiderivatives of the function. It’s essential because differentiating any of these antiderivatives yields the original function.

**Q5:** How does integration by parts work?

**Answer: **Integration by parts, based on the product rule of differentiation, is used for integrals of the form $\int u \, dv$. It follows the formula $\int u \, dv = uv – \int v \, du$. The choice of $u$ and $dv$ is crucial and often follows the LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) rule for selection.

**Q6:** What are some common mistakes to avoid in integration?

**Answer:** Common mistakes include not applying the chain rule correctly in substitution, errors in algebraic simplification, incorrect bounds in definite integrals, and forgetting the constant of integration in indefinite integrals.

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