Integration by parts examples, Single Variable Calculus

We take a quick look at integration by parts (and its derivation), then work through four examples. One of the examples finds the antiderivative of the natural log function ln(x). The method of integration by parts deals with the integral of the product of two functions. We identify one function as f(x) and the other as g'(x). The function f(x) should be straightforward to differentiate, and g'(x) should be one that is similarly simple to integrate. The definitive integral is then transformed into a product of f(x) and g(x), evaluated from a to b, minus the integral of f'(x) and g(x). For clarity and brevity in calculations, we adopt a preferred notation: u for what we will differentiate and dv for what we will anti-differentiate. This leads to the integral of u and dv being expressed as uv evaluated from a to b, minus the integral of v and du: ∫ u dv = uv - ∫ v du. The derivation of this method is a direct application of the product rule for derivatives. By rearranging the terms of the product rule and integrating, we arrive at this formula. Throughout these problems, the aim is not only to perform the integration but to demonstrate a logical approach to choosing u and dv. #calculus #mathematics #integration #integrationbyparts #integralcalculus #mathtutorial #integrationtechniques #calculus1 #calculus2