Integration: Integration by Parts

I am trying to integrate e to a variable power times sin or cos using integration by parts, but I seem to be going in circles. How is this problem solved? The trick for solving e times sin or cos is shown using the example Se^x*sinxdx.

I am trying to integrate a natural log standing alone. How is that done?

The steps for integrating an ln standing alone are shown using the example Slnxdx. The same procedure can also be used for integrals of lns that can be simplified using the properties of logs such as ln3x, ln(x^2) or ln(square root of x), or if the entire ln is raised to a power.

Newton's Law of Cooling relating to differential equations.

At 10:00 AM, an object is removed from a furnace and placed in an environment with a constant temperature of 68 degrees. Its core temperature is 1600 degrees. At 11:00 AM, its core temperature is 1090 degrees. Find its core temperature at 5:00 PM on the same day.

differential equation from calculus II

Given the differential equation: (y^4)(e^2x) + y' = 0 NOTE: The differential equation above is attached in a microsoft word document for better legibility. Additionally my work is attached as a jpeg file. The questions: a)Find the general solution. b)Find the particular solution such that y(0) = 1.

Working with inverse functions and their derivatives.

Given the function f(x) = e^(2x) a) Find the derivative b) Find the inverse (i.e. g(x)) c) Find the derivative of the inverse d) Find the value of g'(pi)

How do I integrate sine to an odd power of 3 or higher?

The steps for integrating sine to an odd power of 3 or higher are shown using the example Ssin^5(x)dx. The solution is detailed and well presented.

How do I integrate cosine to an odd power of 3 or higher?

The steps for integrating cosine to an odd power of 3 or higher are shown using the example Scos^5(x)dx.

How do I integrate sec^3(x)? (or sec to a higher odd power)

The steps for integrating sec^3(x) are shown with explanations.

How do I integrate sin or cos to an even power greater than 2?

The steps for integrating sin or cos to an even power greater than 2 are shown using the example Ssin^4(x)dx.

Functions : Critical Values

The function f(x)=2x^3 - 33x^2 + 108x - 6 has two critical numbers. The smaller one equals ______ and the larger one equals______.