Deepak Sharma

You’ll find that there are many ways to solve an integration problem in calculus. The following list contains some handy points to remember when using different integration techniques: Guess and Check. This technique works when the integrand is close to a simple backward derivative. u-substitution. The integration counterpart to the chain rule; use this technique when the argument of the function you’re integrating is more than a simple x. Integration by Parts. Integration’s counterpart to the product rule. 1. Use this technique when the integrand contains a product of functions. 2. Pick your u according to LIATE, box it, “7” it, finish it. Trig Integrals 1. Use Pythagorean identities. 2. Use half-angle formula Trigonometric Substitution. This method works when the integrand contains radicals of the forms (or powers of these roots), where a is a constant and u is an expression in x. Partial Fractions. This technique works for rational functions (one polynomial over another).

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.) The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative, the doubling function.

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