Integral of Logarithmic Functions | Calculus

This paragraph introduces the concept of finding the indefinite integral of logarithmic functions. It presents a formula for integrating logarithmic functions, which is the integral of log base a of u, du, equated to u times log base a of u divided by e, with e being the constant 2.7182818. The paragraph emphasizes that this formula applies when u is a linear function, such as 5x, x, or 3x plus 4, but not when u is a nonlinear function like x cubed plus 9. The example provided involves integrating log base 4 of x and demonstrates how to apply the formula to find the answer, which involves the constant of integration.

This paragraph delves deeper into the process of integrating logarithmic functions by showing the steps to solve the original problem of finding the integral of log base 4 of x. It explains the use of the change of base formula, converting the logarithm to base e (natural log), and then applying integration by parts. The paragraph outlines the integration process, which involves making appropriate choices for u and dv, and using the integral of u dv formula. It then demonstrates how to simplify the expression to obtain the final answer, which in this case is x log base 4 x over e plus a constant of integration.

The paragraph presents a more complex example of integrating a logarithmic function, specifically the integral of log base 5 of x plus 7. It guides through applying the formula for integrating logarithmic functions when u is a linear function, which in this case is x plus 7. The explanation includes using the change of base formula, integration by parts, and u substitution to simplify the integral. The paragraph concludes with the final answer, which is expressed as a combination of logarithmic expressions and constants, and encourages the viewer to follow along and practice the steps.

This paragraph focuses on the factoring and simplification of logarithmic integrals. It corrects a previous mistake and emphasizes the importance of factoring out the log expression. The paragraph demonstrates how to simplify the expression by combining logarithms and using the change of base formula to convert back to the original logarithmic form. The final answer is presented in a factored form that highlights the log base 5 of the given expression, minus log base 5 of e, plus a constant.

The paragraph tackles another problem involving the integral of log base 3 of 5x plus 9. It begins by applying the formula for integrating logarithmic functions and then proceeds to use the change of base formula to convert the problem into a more manageable form. The explanation includes defining u and v for integration by parts and solving the integral using u substitution. The paragraph concludes with the final answer, which is a combination of logarithmic expressions and constants, and encourages the viewer to understand the step-by-step process to reach the solution.

This paragraph discusses the approach for integrating logarithmic expressions that involve exponents and perfect square trinomials. It explains how to move the exponent to the front when dealing with logs and how to factor perfect square trinomials. The paragraph provides two examples: one with an exponent in the logarithm and another with a perfect square trinomial. It shows how to simplify these expressions using the properties of logs and the change of base formula, resulting in the final answers in the form of integrals of logarithmic functions.

The final paragraph addresses the challenge of integrating logarithmic functions when the function u is not linear. It suggests strategies for dealing with such cases, such as moving exponents to the front for logarithmic expressions and factoring perfect square trinomials. The paragraph emphasizes that if neither of these strategies can be applied, the problem becomes more difficult. It concludes by encouraging viewers to be aware of these approaches when faced with complex logarithmic integrals.