Increasing functions: Comparing Functions

Explain why the graph of f(x) is rising over an interval a < or equal to x < or equal to b if f '(x) > 0 throughout the interval. What can you say about the graph of f if f '(x) is less than zero on a < or equal to x < or equal to b?

U substitution

U substitution is explained using the example S4x(x^2+1)^5dx without and with limits of integration.

Integrate sin^2(x)?

The steps for integrating sin^2(x) are shown with explanations of each step.

Tabular integration

I am integrating x to a power times sin, cos or e to a power using repeated integration by parts, but it is very cumbersome. Is there a better way?

How do I integrate cos^2(x)?

The steps for integrating cos^2(x) are shown with explanations.

How do I integrate tan^2(x)?

The steps for integrating tan^2(x) are shown with explanations.

How do I integrate cot^2(x)?

The steps for integrating cot^2(x) are shown with explanations.

Find a function whose derivative exists and is not continuous

Find a real valued function such that its derivative exists in every point, but it is not continuous at least in one point.

U substitution

The steps of U substitution are explained using the example S6/(2x+1)^4dx.

Maximization

A Farmer wants to construct a fence. The area that he is going to enclose is rectangular, but one of the sides is a river (assumed to be straight). If he has 120 m of fence, what is the maximum surface that he can enclose?