A power function is a mathematical expression that involves raising a variable to a fixed exponent, while a trigonometric function is a mathematical function that relates the angles of a triangle to the lengths of its sides. In other words, power functions involve only algebraic operations, while trigonometric functions involve trigonometric ratios such as sine, cosine, and tangent.
An integral of a power function is the reverse process of differentiation. It represents the area under the curve of a power function and can be calculated using the fundamental theorem of calculus. The integral of a power function with exponent n is given by (x^(n+1))/(n+1) + C, where C is a constant.
To integrate a trigonometric function, you need to use trigonometric identities and substitution. By substituting the variable with a trigonometric function and using trigonometric identities, the integral can be simplified into a form that can be integrated using basic integration techniques.
The purpose of finding the integral of a function is to calculate the area under the curve of the function. This can be useful in many real-life applications, such as calculating the work done by a force, finding the displacement of an object, or determining the total cost of a product based on its rate of change.
No, not all functions can be integrated. Some functions are not continuous or have discontinuities, making it impossible to find their integral. However, most elementary functions can be integrated using various integration techniques.