Integration

Understanding Basic Integration

• Integration is the reverse process of differentiation, often used to find the original function given its derivative.
• The integral sign (∫) is used to represent integration in an equation.
• For any power of x, the basic rule is that you increase the power by 1 and divide by the new power.

Example: ∫x^n dx = (1/n+1) * x^(n+1) + C

Definite and Indefinite Integrals

• An indefinite integral is a family of functions that includes an arbitrary constant (denoted by C); it does not have limits.
• A definite integral calculates the net area bounded by a function and the x-axis between two specified points.

Basic Functions Integrals

• Understanding the integrals of basic functions is crucial. The following are highly important:
  • ∫dx = x + C
  • ∫x dx = 0.5x^2 + C
  • ∫sinx dx = -cosx + C
  • ∫cosx dx = sinx + C

Integration For Area Under a Curve

• Integration can be used to calculate the area under a curve from a certain point a to point b.
• The area under the curve is given by the definite integral ∫ from a to b of f(x) dx.

Applications of Integration

• Integration is used in various fields such as physics, economics, and engineering.
• Applications include motion analysis (position, velocity, acceleration), finding areas and volumes, solving differential equations, and in probability and statistics.

Understanding the techniques and applications of integration is crucial to solve various mathematical problems. Practice is key, so work through lots of examples to apply and reinforce these concepts.