Techniques of integration from "summary" of Differential Calculus by S Balachandra Rao
The techniques of integration serve as valuable tools in the realm of calculus, enabling us to find antiderivatives and calculate definite integrals effectively. Through a series of methods and strategies, we are able to tackle a wide array of functions and successfully integrate them. These techniques provide us with the means to solve complex problems and gain a deeper understanding of the behavior of functions. One such technique is integration by substitution, where we substitute a new variable in place of the original variable in order to simplify the integrand. This method proves to be particularly useful when dealing with composite functions or expressions that are not easily integrable. By making the appropriate substitution, we are able to transform the integral into a more manageable form, allowing us to solve it with ease. Another important technique is integration by parts, which is based on the product rule for differentiation. This method involves choosing two functions to differentiate and integrate, respectively, in order to simplify the original integral. By applying the integration by parts formula, we are able to express the integral as a product of these two functions, making it easier to evaluate. Furthermore, trigonometric substitutions are commonly used when dealing with integrals involving trigonometric functions. By making the appropriate substitution based on trigonometric identities, we are able to simplify the integral and express it in terms of elementary functions. This technique proves to be especially helpful in solving integrals that involve trigonometric expressions. Additionally, partial fractions decomposition is a useful technique for simplifying rational functions by breaking them down into simpler fractions. By decomposing the original function into partial fractions, we are able to express it as a sum of simpler integrals, making it easier to evaluate. This method is particularly handy when dealing with rational functions that are not easily integrable.- The techniques of integration play a crucial role in the field of calculus, providing us with the tools needed to solve a wide range of integrals effectively. By employing methods such as integration by substitution, integration by parts, trigonometric substitutions, and partial fractions decomposition, we are able to tackle complex integrals and gain a deeper understanding of the behavior of functions. Through these techniques, we are able to navigate the intricacies of calculus and apply our knowledge to solve challenging problems.
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