# Calculus

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### Integrals of Ln(e^x + 1) and Ln(e^x - 1)

Ln(e^x + 1) and Ln(e^x - 1) are non-integrable functions (no closed-form antiderivatives), but their integrals can be approximated with infinite series.

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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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### 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 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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### Integral of Ln(1 + x^3) and Ln(x^3 - 1)

How to integrate ln(x^3 + 1), natural logarithm of cubic polynomial. Use polynomial factorization to simplify logarithm functions and find their antiderivatives...

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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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### Top 4 Mistakes Calculus Students Make with u Substitution

It can be your friend or you foe when trying to solve integrals, the sometimes dreaded and often misunderstood technique of u substitution. In integral calculus, substituting a function of u for x in the integral ...

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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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### Sample Calculus 1 Exam Questions

min - max problems on differential calculus exams, using first derivatives to solve calculus test questions, examples and practice test problems for Calc I students in high school or college

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