What is the integral of a function?

Integration is the process of finding the total accumulation or area under the curve of a function.

What is the fundamental theorem of calculus?

It states that differentiation and integration are inverse operations, connecting the two through the relationship between a function and its antiderivative.

What distinguishes an indefinite integral from a definite integral?

Indefinite Integral: Represents the family of all antiderivatives, includes a constant C. Definite Integral: Calculates the total accumulation of the area under the curve over a specific interval.

What is integration by substitution?

It’s a method used to simplify an integral by substituting a part of the integrand with a new variable.

How do differentiation and integration relate to each other?

Differentiation and integration are inverse operations: differentiation finds the rate of change, and integration finds the total accumulation.

Basics of Differentiation and Integration Explained

Q: What is differentiation?

Differentiation is the process of finding the rate of change or the slope of a function at a given point.

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Differentiation and Integration

Calculus is a branch of mathematics that focuses on change and motion. Two fundamental operations in calculus are differentiation and integration, which help to describe how things change and accumulate. These two concepts are not just essential in mathematics but also have practical applications in physics, economics, engineering, and even biology.

1.0Basics of Differentiation and Integration

At the core of calculus, differentiation and integration are inverse operations, each with its specific purpose:

Differentiation: It is the process of finding the rate of change or the slope of a function at any point. In simple terms, differentiation tells you how a quantity changes as the input changes. The result of differentiation is called a derivative.

Integration: It is the reverse of differentiation. Integration is the process of finding the total accumulation of a quantity or the area under a curve. The result of integration is called an integral.

Together, these operations allow mathematicians and scientists to model and understand change and accumulation in the real world.

2.0Derivation and Integration

Derivation (Differentiation) refers to the process of determining the rate of change of a function. For example, if you have a position-time function that represents the motion of an object, the derivative of this function will give you the velocity (the rate of change of position).
Integration on the other hand, helps in finding the accumulated value. Using the example of motion, if you know the velocity-time function, the integral of that function gives you the position of the object.

3.0Differentiation and Integration Examples

1. Differentiation Example: Differentiate $x^{2}$

Solution:

f'(x) = 2x

Explanation: The derivative of x^2 is 2x, which tells you how the value of the function changes as x changes.

2. Integration Example: Integrate f(x) = 2x

Solution:

$\int 2 x d x = x^{2} + C$

Explanation: The integral of 2x with respect to x gives $x^{2} + C,$ where C is the constant of integration.

4.0Differentiation and Integration of Trigonometric Functions

Both differentiation and integration are applicable to trigonometric functions, and the rules differ slightly for each.

Differentiation	Integration
$\frac{d}{d x} (sin x) = cos x$	$\int cos x d x = sin x + C$
$\frac{d}{d x} (cos x) = - sin x$	$\int sin x d x = - cos x + C$
$\frac{d}{d x} (tan x) = sec^{2} x$	$\int sec^{2} x d x = tan x + C$
$\frac{d}{d x} (cot x) = - csc^{2} x$	$\int sec x tan x d x = sec x + C$
$\frac{d}{d x} (sec x) = sec x tan x$	$\int csc x cot x d x = csc x + C$
$\frac{d}{d x} (csc x) = - csc x cot x$	$\int csc^{2} x d x = - cot x + C$

5.0Integration and Differentiation in Calculus

In calculus, differentiation and integration play complementary roles. Differentiation breaks down a function into its rate of change, while integration provides the means to accumulate those changes. This connection between the two operations is formally described by the Fundamental Theorem of Calculus, which states:

If a function f(x) is continuous on an interval and F(x) is an antiderivative of f(x), then the integral of f(x) from a to b is given by:

$\int_{a}^{b} f (x) d x = F (b) - F (a)$

This theorem demonstrates that differentiation and integration are inverses of each other, providing a powerful way to solve real-world problems involving motion, growth, and area.

6.0Integration and Differentiation Rules

There are a few essential rules and formulas that govern both differentiation and integration. Let’s look at some of the most important ones:

Differentiation Rules

Rule	Function	Differentiation
Power Rule:	$f (x) = x^{n}$	$(f^{'} (x) = n . x^{n - 1}$
Sum Rule:	$f (x) = g (x) + h (x)$	$f^{'} (x) = g^{'} (x) + h^{'} (x)$
Difference Rule:	$f (x) = g (x) - h (x)$	$f^{'} (x) = g^{'} (x) - h^{'} (x)$
Product Rule:	$f (x) = g (x) \cdot h (x)$	$f^{'} (x) = g^{'} (x) \cdot h (x) + g (x) \cdot h^{'} (x)$
Quotient Rule:	$f (x) = \frac{g ( x )}{h ( x )}$	$f^{'} (x) = \frac{g ^{'} ( x ) \cdot h ( x ) - g ( x ) \cdot h ^{'} ( x )}{h ( x ) ^{2}}$
Chain Rule:	$f (x) = g (h (x))$	$f^{'} (x) = g^{'} (h (x)) \cdot h^{'} (x)$

Integration Rules

Rule	Function	Integration
Power Rule:	$f (x) = x^{n}$	$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$
Sum Rule:	$f (x) = g (x) + h (x)$	$\int f (x) d x = \int g (x) d x + \int h (x) d x$
Difference Rule:	$f (x) = g (x) - h (x)$	$\int f (x) d x = \int g (x) d x - \int h (x) d x$
Integration by Substitution:	$f (x) = g (h (x)) \cdot h^{'} (x)$	$\int f (x) d x = \int g (u) d u where u = h (x)$
Integration by Parts:	$\int u d v$	$= uv - \int v d u$

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