promath The limit of x^1/x as x approaches infinity is equal to 1. This limit is denoted by limx→∞ x1/x, so the formula for the limit of x1/x when x tends to infinity is given by limx→∞ x1/x = 1. Let us now find the limit of x to the power 1/x when x tends to … Read more
The Laplace transform of e^4t is equal to 1/(s-4) and the Laplace of e^-4t is equal to 1/(s+4). This is because, we know that the Laplace of eat is 1/(s-a). The Laplace transform formulae for the functions e4t and e-4t are given in the table below. Function f(t) L{ f(t) } e4t L{e4t} = $\dfrac{1}{s-4}$ … Read more
The Laplace transform of e^3t is equal to 1/(s-3) and the Laplace of e^-3t is equal to 1/(s+3). This is because, we know that the Laplace of eat is 1/(s-a). The Laplace transform formula for the functions e3t and e-3t are given as follows. Laplace of e3t We will find the Laplace transform of e3t … Read more
The Laplace transform of e^2t is equal to 1/(s-2) and the Laplace of e^-2t is equal to 1/(s+2). More generally, the Laplace of eat is 1/(s-a). The Laplace formula for the functions e2t and e-2t are given below: Laplace of e2t By definition, the Laplace of f(t) is given by the integral L{f(t)} = $\int_0^\infty$ … Read more
The Laplace transform of sin2t is equal to L{sin2t} = 2/(s2+4) and the Laplace of cos2t is L{cos2t} = s/(s2+4). Because, the Laplace of sinat is a/(s2+a2) and the Laplace of cosat is s/(s2+a2). In this post, we will find the Laplace transform of sin2t and cos2t. Laplace of sin2t and cos2t To find the … Read more
The limit of (1-cosx)/x^2 as x approaches 0 is equal to 1/2. That is, limx→0 (1-cosx)/x2 =1/2. This limit can be computed using the formula limx→0 $\dfrac{\sin x}{x}$ = 1 …(∗) Prove that limx→0 (1-cosx)/x2 =1/2 We have limx→0 $\dfrac{1-\cos x}{x^2}$ = limx→0 $\dfrac{2 \sin^2 \frac{x}{2}}{x^2}$ using the formula 1-cos2x=2sin2x. = limx→0 $\dfrac{2 \sin^2 \frac{x}{2}}{(\frac{x}{2})^2 … Read more
The derivative of sin(xy) is equal to (y+x dy/dx) cos(xy), and this is the derivative of sin(xy) with respect to x. The derivative of sin(xy) formula is given below: $\dfrac{d}{dx}(\sin xy)=(y+x\dfrac{dy}{dx})\cos xy$. Differentiate sin(xy) with respect to x Answer: The derivative of sin(xy) with respect to x is equal to (y+x dy/dx) cos(xy). Explanation: Let … Read more
If y=cos(x+y), then dy/dx= -sin(x+y)/[1+sin(x+y)]. Here, we learn how to differentiate y=cos(x+y) with respect to x. Let us find the derivative of y=cos(x+y). Find dy/dx if y=cos(x+y) Question: If y=cos(x+y), then $\dfrac{dy}{dx}$. Solution: We are given that y = cos(x+y). Step 1: Differentiating both sides of the above equation with respect x, we get that … Read more
If y=sin(x+y), then dy/dx= cos(x+y)/[1-cos(x+y)]. Here, we learn how to differentiate y=sin(x+y) with respect to x. Let us find the derivative of y=sin(x+y). y=sin(x+y), Find dy/dx Question: If y=sin(x+y), then $\dfrac{dy}{dx}$. Solution: Given, y = sin(x+y). To find dy/dx, we will differentiate both sides of the equation y=sin(x+y) with respect x. Using the chain rule, … Read more
The integral of cos root x dx is denoted by ∫cos(√x)dx, and it is equal to ∫cos(√x)dx = 2[√xsin(√x)+ cos(√x)]+C where C is an integration constant. Here we will learn how to integrate cos root x. The integral formula of cos root x is given below. $\int \cos \sqrt{x} dx$ $= 2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]+C$ … Read more