The Indefinite Integral

The Indefinite Integral

The indefinite integral, also known as the antiderivative, is the reverse process of differentiation. It helps in finding original functions when given their derivatives and is fundamental in solving differential equations, physics, and engineering problems.

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1. Definition of the Indefinite Integral

An indefinite integral of a function $f(x)$ is written as:

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

where:

• $F(x)$ is the antiderivative of $f(x)$, meaning $F'(x) = f(x)$.
• $C$ is the constant of integration, which accounts for any constant lost during differentiation.

For example:

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

because the derivative of $x^2 + C$ is $2x$.

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2. Basic Rules of Indefinite Integration

1. Power Rule

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

Example:

$$\int x^3 \, dx = \frac{x^4}{4} + C$$

2. Constant Rule

$$\int k \, dx = kx + C$$

Example:

$$\int 5 \, dx = 5x + C$$

3. Constant Multiple Rule

$$\int k f(x) \, dx = k \int f(x) \, dx$$

Example:

$$\int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \cdot \frac{x^3}{3} + C = x^3 + C$$

4. Sum/Difference Rule

$$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$$

Example:

$$\int (x^2 + 3x) \, dx = \int x^2 \, dx + \int 3x \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + C$$

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3. Common Indefinite Integrals

Function $f(x)$	Indefinite Integral $\int f(x) \, dx$
$x^n$ (for $n \neq -1$)	$\frac{x^{n+1}}{n+1} + C$
$e^x$	$e^x + C$
$\ln x$	$x \ln x - x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$
$\sec x \tan x$	$\sec x + C$
$\csc x \cot x$	$-\csc x + C$

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4. Example Problems

Example 1: Compute $\int (3x^2 + 2x - 5) \, dx$

Using the sum and power rules:

$$\int 3x^2 \, dx + \int 2x \, dx - \int 5 \, dx$$ $$= 3 \cdot \frac{x^3}{3} + 2 \cdot \frac{x^2}{2} - 5x + C$$ $$= x^3 + x^2 - 5x + C$$

Example 2: Compute $\int e^x + \frac{1}{x} \, dx$

Using the table of integrals:

$$\int e^x \, dx + \int \frac{1}{x} \, dx$$ $$= e^x + \ln |x| + C$$

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5. Applications of Indefinite Integrals

• Physics: Finding velocity from acceleration, displacement from velocity.
• Engineering: Solving differential equations in circuits and mechanics.
• Economics: Computing demand and supply functions.
• Probability: Determining cumulative distribution functions.

The indefinite integral is essential for solving problems where the original function needs to be reconstructed from its rate of change.