Calculus

Maclaurin series of cosx | Taylor series of cosx

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The Maclaurin series expansion of cosx or the Taylor series expansion of cosx at x=0 is given as follows: cosx = $\sum_{n=0}^\infty \dfrac{(-1)^n}{(2n)!}x^{2n}$ = $1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\cdots$ Taylor Series Expansion of Sinx at x=0 Note that the Maclaurin series expansion of f(x)=cosx or the Taylor series of a function f(x) at $x=0$ is given by the following … Read more

Limit of x/sinx as x approaches 0

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The value of the limit of x/sinx as x approaches 0 is equal to 1. In this post, we will learn how to find the limit of $\frac{x}{\sin x}$ when x tends to 0. lim x/sinx when x approaches 0 Formula The formula of the limit of x/sinx when x tends to zero is given … Read more

Integral of e^(x^2) | Integration of e^(x^2)

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 The integral of e^(x^2), that is, the integration of e to the x2 is equal to $\dfrac{\sqrt{\pi}}{2}$ erfi(x)+C, where erfi(x) is called the “imaginary error function” and C is an integration constant. In this post, we will learn to integrate the function $e^{x^2}$. Integral of $e^{x^2}$ We know that ex can be written as ex … Read more

Integral of e^2x | e^2x Integration

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The integration of e2x is equal to e2x/2. In this post, we will learn how to integrate e to the 2x. The integration of e2x can be derived directly from the formula of the integral of emx which is given below: ∫emx dx = $\dfrac{e^{mx}}{m}$ +C where C is an integration constant. Putting m=2 in … Read more

Integration of tanx: Formula, Proof | tanx Integration

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The integration of tanx is -ln|cosx| or ln|secx|, where ln denotes the natural logarithm, that is, logarithm with base e. Here we will learn how to find the integral of tanx dx. tanx Integration Formula The tanx integration formula is given below. Integration of tanx Proof We will show that ∫tanx dx = -ln|cosx|+C. As … Read more

How to Integrate Root (a^2+x^2) | Integration of Root(a^2+x^2)

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The integration of the square root of a2+x2 is given as follows: $\int \sqrt{a^2+x^2} dx$ $=\dfrac{x}{2} \sqrt{a^2+x^2}$ $+\dfrac{a^2}{2}\log |x+\sqrt{a^2+x^2}|+C$, where $C$ is an integration constant. In this post, we will find the integral of root(a2+x2). Let’s learn how to integrate square root of a2+x2. Integration of $\sqrt{a^2+x^2}$ Question: What is the integration of square root … Read more

Derivative of cos2x by First Principle, Chain Rule

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The derivative of cos2x is equal to -2sin2x.  In this post, we will find the derivative of cos2x by the first principle, that is, by the limit definition of derivatives as well as by the chain rule of derivatives. Recall the first principle of derivatives. By this rule, we know that the derivative of a … Read more

Derivative of 3^x by First Principle

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The derivative of 3x is equal to 3x ln 3. Here ln 3 denotes the natural logarithm of 3, that is, ln 3 = loge 3. In this post, we will find the derivative of 3 to the power x by the first principle of derivatives. We first recall the first principle of derivatives. The … Read more