# Calculus

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### How to Integrate Sqrt(e^x + 1) and Sqrt(e^x - 1)

The functions y = sqrt(e^x + 1) and sqrt(e^x - 1) are very similar in appearance and behavior to the function sqrt(e^x) = e^(x/2). On the interval [1, ∞) they grow like exponential functions, as you can see in the...

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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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### 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 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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### How to Find the Derivative of a Simple Rational Function (ax + b)/(x + c)

A rational function is a fraction with polynomials as the numerator and denominator. The simplest rational function is a ratio of two linear functions, (ax + b)/(x + c). Such functions are often encountered in applied...

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### How to Integrate Tan(x)^4 and Tan(x)^5 -- Tan^4(x) and Tan^5(x)

The fourth and fifth powers of the tangent function can be written either as tan(x)^4 and tan(x)^5, or as tan^4(x) and tan^5(x). To integrate these functions, you need to apply trigonometric identities and use either...

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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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### How to Integrate Arctan(Sqrt(X))

Integrals involving trig functions, inverse trig functions, and square roots often do not have closed form expressions. For example, the integrals ∫ cos(sqrt(x)) dx ∫ sqrt(x)*sin(x) dx ∫ sqrt(arctan(x)) dx...

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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 the Functions x*sin(x) and x*cos(x)

Trigonometric product functions such as x*sin(x) and x*cos(x) are often encountered in physics in wave or oscillation problems, or as solutions to linear differential equations. Finding the antiderivative of either...

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