# Calculus

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### How to Integrate Sine Squared and Cosine Squared: sin(x)^2 and cos(x)^2

The integrals of sin(x)^2 and cos(x)^2 [also written sin^2 (x) and cos^2 (x)] are found by using trigonometric identities, in particular, the double-angle formula. Transforming sine squared and cosine squared into...

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### Antiderivative of Sqrt(Cos(x)) and Sqrt(Sin(x))

Though they look simple, the square root of cosine and square root of sine are non-integrable functions, meaning their antiderivatives cannot be expressed in closed-form in terms of elementary functions. No matter how...

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### Differential Equations, part 1: Elimination of Arbitrary Constants

Going to heaven? begins at par. 10. DEs finds functions. This hub works backwards. You are given a function, and you find a differential equation associated with it.

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### Maximize the Viewing Angle | Calculus Optimization Problem

A classic problem in differential calculus is to find the optimal distance from a raised screen so that the viewing angle is maximized. The angle formed by the bottom of the screen, eye, and top of the screen will vary...

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### How to Integrate Sec(x)^3 and Sec(x)Tan(x)^2 -- Trig Antiderivatives

Graph of y = sec(x)^3. Finding the antiderivatives of f(x) = sec(x)^3 and g(x) = tan(x)^2*sec(x) are challenging problems in integral calculus because standard tricks like integration by parts, substitution, and using...

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### How to Integrate y with respect to x in Calculus

Knowing how to integrate y with respect to x using calculus will help you to solve math problems and physics problems. Geometrical problems and problems about moving objects can be solved by integrating y with respect...

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### How to Integrate e^(e^x)

The single-variable function f(x) = e^(e^x) is one of the simplest examples of a non-integrable function. In fact, any double-exponential function of the form f(x) = A^(B^x) is also non-integrable. A non-integrable...

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### How to Maximize the Volume (Capacity) of a Rectangular Box Cut From a Rectangle

A classic problem in calculus is the find the dimensions of a box with maximal volume that can be constructed from a rectangular piece of paper or stiff cardstock. The box is made by cutting four equally sized squares...

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### How to Integrate Sqrt(Tan(x)) | Hard Antiderivatives

In general, the square root of a trig function is not integrable, for example, sqrt(sin(x)) and sqrt(cos(x)) are functions whose antiderivatves cannot be expressed in closed form. However, there are some notable...

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### Integration by Parts: How to Find the Antiderivative of Ln(x)

Integration by parts is a technique for finding the antiderivative of functions that are the product of two functions, usually two simpler functions. It is the analogue of the product rule for derivatives. Given an...

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