# Further integrations

## 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.