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 techni...
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