Antiderivatives and Initial Value Problems

Antiderivatives

Definition and Basic Examples

An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). The process of finding antiderivatives is called integration. The general antiderivative includes a constant of integration, C, because the derivative of a constant is zero.

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

• Example: $\int \frac{1}{x}\,dx = \ln|x| + C$

• Example: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$

Common Antiderivative Formulas

Some basic antiderivatives are essential for solving integrals:

• $\int \sin x\,dx = -\cos x + C$

• $\int \cos x\,dx = \sin x + C$

• $\int e^x\,dx = e^x + C$

• $\int a^x\,dx = \frac{a^x}{\ln a} + C$ for $a > 0, a \neq 1$

• $\int \sec^2 x\,dx = \tan x + C$

• $\int \csc^2 x\,dx = -\cot x + C$

• $\int \sec x \tan x\,dx = \sec x + C$

• $\int \csc x \cot x\,dx = -\csc x + C$

Properties of Indefinite Integrals

Linearity and Constants

Indefinite integrals have several important properties that simplify calculations:

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

• $\int k f(x)\,dx = k \int f(x)\,dx$ for any constant $k$

• $\int 0\,dx = C$

Examples of Evaluating Integrals

To evaluate an indefinite integral, apply the relevant formulas and properties:

• Example: $\int (2\cos x + 3e^x)\,dx = 2\int \cos x\,dx + 3\int e^x\,dx = 2\sin x + 3e^x + C$

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

Initial Value Problems

Solving Differential Equations with Initial Conditions

An initial value problem involves finding a specific solution to a differential equation that satisfies a given initial condition. The general steps are:

1. Integrate the differential equation to find the general solution (including C).

2. Substitute the initial condition to solve for C.

• Example: If $\frac{dy}{dx} = 2x$ and $y(0) = 3$, then $y = x^2 + C$. Plug in $x = 0$, $y = 3$ to get $C = 3$, so $y = x^2 + 3$.

Applications: Position, Velocity, and Acceleration

Relating Motion to Antiderivatives

In physics, the position, velocity, and acceleration of an object are related through derivatives and antiderivatives:

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• To find position from acceleration, integrate twice, applying initial conditions for velocity and position.

• Example: If $a(t) = -32$ (gravity, in ft/s2), $v(0) = 10$, $s(0) = 5$:

• Integrate $a(t)$: $v(t) = -32t + C_1$

• Use $v(0) = 10$ to find $C_1 = 10$

• Integrate $v(t)$: $s(t) = -16t^2 + 10t + C_2$

• Use $s(0) = 5$ to find $C_2 = 5$

• Final position function: $s(t) = -16t^2 + 10t + 5$

Summary Table: Common Antiderivatives

Function $f(x)$	Antiderivative $F(x)$
$x^n$ ($n \neq -1$)	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln|x| + C$
$e^x$	$e^x + C$
$a^x$	$\frac{a^x}{\ln a} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$
$\sec x \tan x$	$\sec x + C$
$\csc x \cot x$	$-\csc x + C$