Further integrations

Fundamentals of Further Integration

Integration is the process of finding the integral of a function, which often represents an area under a curve or a total quantity from a rate of change.
The fundamental theorem of calculus states that differentiation and integration are opposite processes.
An indefinite integral is a family of functions F that differentiate to give the original function. Written as ∫f(x) dx = F(x) + c.
A definite integral is the net area under a curve between two points a and b. Written as ∫ (from a to b) of f(x) dx.
The power rule for integration is the reverse of the power rule for differentiation, ∫x^n dx = (1/(n+1))x^(n+1) + c, for n ≠ -1.

Methods of Integration

Substitution is used when the integral contains a function and its derivative. The substitution u = g(x) simplifies the integral to the form ∫f(u) du.
Integration by parts, based on the product rule for differentiation, is used for integrals of the form ∫u v dx. It’s often used when a product of functions is involved with one part being easily integrable and the other having a simpler derivative.
Partial fractions are used when integrating rational functions (where the numerator has a lower degree than the denominator). By expressing the fraction as simpler fractions, more straightforward integrals are created.

Integration of Trigonometric, Exponential, and Logarithmic Functions

Trigonometric functions can be integrated using Pythagorean identities and trigonometric substitution.
The integrals of sin(x), cos(x), sec^2(x), and csc^2(x) are related to readily familiar functions.
The integral of e^x is itself, while for a^x (a > 0, a ≠ 1), the integral is (a^x)/log(a).
For logarithmic functions, integral of ln x is xln x - x.

Applying Integration in Practical Contexts

Integration can be used for finding areas between curves, by taking the difference between the integrals of two functions.
It is also used in solving differential equations, particularly in physics, engineering and economics.
The mean value theorem for integrals and the first and second fundamental theorems of calculus are also significant areas of application.
One common practical use is in calculating work done, with the integral of force over distance.
It can also be used for calculating volume using disk method or shell method.

Numerical Integration

Sometimes it’s impossible or too difficult to find the exact area under a curve. In these instances, numerical integration, including the trapezium rule, midpoint rule or Simpson’s rule, can provide an approximation.