# Calculus

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### How to Integrate 1/(x^4 + 1) by Hand

Antiderivatives of rational functions become more and more complicated as the degree of the denominator polynomial increases. We have already seen in other tutorials how to find the integrals of 1/(ax^2 + bx + c),...

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### Hanging Cable Problems | Catenary Math

Equations for catenaries given the distance between the ends of the hanging rope, the length of the rope, or the length of sag. How to solve problems with hanging cables, chains, or ropes.

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### Why Does d/dx[Sin(x)] = Cos(x)? Complete Proof of the Derivative of Sin(x)

Proving that the derivative of sin(x) equals cos(x) -- using nothing but the limit definition of the derivative and basic trigonometric and geometric principles -- is rather involved and requires some clever tricks that...

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### How to Solve y" = f(y), Second-Order Non-Linear Differential Equations

Differential equations of the form y" = f(y), or d2y/dx2 = f(y) where y is function of x and f is a non-linear function, are examples of non-linear second-order ordinary differential equations. As this equation...

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### How to Integrate x(e^x)cos(x) and x(e^x)sin(x)

The method of integration by parts can be used to find the antiderivatives of two-product functions such as x*cos(x), x*e^x, and (e^x)cos(x), but what about the triple product of a polynomial, trig function and...

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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(1 + Cos(x)) and Similar Functions

Graph of y = sqrt(1 + cos(x)). Finding the integral of sqrt(1 + cos(x)), sqrt(1 - cos(x)), sqrt(1 + sin(x)), sqrt(1 - sin(x)) and similar functions may seem challenging since there is no obvious way to apply a...

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### How to Integrate sin(x^2) and cos(x^2)

The functions f(x) = sin(x^2) and g(x) = cos(x^2) are examples of non-integrable functions, that is, functions whose antiderivatives cannot be expressed in terms of elementary functions. They can be expressed in terms...

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### Two Ways to Integrate Sec(x), Secant of x

Graph of sec(x). The curve of the function is shown in red, the vertical asymptotes are shown in pink. In trigonometry, the secant function is the reciprocal of cosine, that is, sec(x) = 1/cos(x) Using the complex...

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### How to Integrate Tan^2(x) and Tan^3(x), Tangent Squared and Cubed

Graphs of f(x) = tan(x)^2 (blue) and g(x) = tan(x)^3 - 1 (orange). Tangent squared and tangent cubed -- written as either tan^2(x) & tan^3(x), or tan(x)^2 & tan(x)^3 -- are easy to integrate if you know a few...

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