Integration by parts and substitution from "summary" of Differential Calculus by S Balachandra Rao
Integration by parts and substitution are essential techniques in calculus that allow us to solve complex integrals by breaking them down into simpler parts. Let us first discuss integration by parts. This method is based on the product rule for differentiation and is used to integrate the product of two functions. The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are functions of x. To apply this formula, we choose u and dv such that we can easily differentiate u and integrate dv. For example, if we have to find the integral of x*sin(x) with respect to x, we can choose u = x and dv = sin(x)dx. Then, we differentiate u to get du = dx and integrate dv to get v = -cos(x). Substituting these values into the formula, we get the integral as -x*cos(x) - ∫-cos(x)dx. Next, let us discuss the method of substitution. This technique involves substituting a new variable or expression in place of the existing variable in the integral. The goal of substitution is to simplify the integral by making it easier to integrate. For example, when faced with integrals involving trigonometric functions, we can use trigonometric identities to simplify the expression before integrating. Similarly, for integrals involving rational functions, we can use a suitable substitution to transform the integral into a simpler form. By combining integration by parts and substitution, we can tackle a wide range of integrals that would otherwise be challenging to solve. These methods require practice and a good understanding of the underlying concepts to apply them effectively. By mastering these techniques, we can solve a variety of integrals and gain a deeper insight into the world of calculus.
Similar Posts
Introduction to RRB ALP Exam
The first step towards cracking the RRB ALP exam is to understand the basics of the exam itself. This examination is conducted ...
Integration plays a key role in the study of sequences and series
The concept of integration is fundamental in understanding sequences and series in mathematics. By applying integration techniq...
Integrals can be evaluated using limits and Riemann sums
To evaluate integrals, we can utilize limits and Riemann sums. This concept is crucial in integral calculus and plays a signifi...
Taylor series expansions can be derived using integration
To understand how Taylor series expansions can be derived using integration, we need to start by recalling the definition of a ...
Multiple integrals allow for the calculation of volumes in higher dimensions
Multiple integrals are a powerful tool in calculus that extend the idea of a single integral to higher dimensions. They are use...
Integration plays a crucial role in probability theory and statistics
Integration plays a crucial role in probability theory and statistics. When dealing with probability distributions, it is often...