promath The Laplace transform of unit step function is equal to 1/s. The unit step function, denoted by, u(t), is defined as follows: $u(t) = \begin{cases} 0, & \text{if } t<0 \\ 1, & \text{if } t \geq 0. \end{cases}$ The Laplace transform of u(t), unit step function, is given by L{u(t)} = $\dfrac{1}{s}$. Laplace transform … Read more
The Laplace transform of e^t is equal to L{et} = 1/(s-1) whenever s>1. Here we find the Laplace of et by the definition of Laplace transforms. The formula of the Laplace of et is given by L{et} = $\dfrac{1}{s-1}$. Find the Laplace Transform of et By definition of Laplace transforms, the Laplace transform of a … Read more
Answer: The Laplace transform of 0 is equal to L{0} = 0. In this post, we learn to find the Laplace of zero. The formula of the Laplace transform of zero is given as follows: L{0} = 0. Laplace Transform of 0 By definition, the Laplace of f(t) is given by the integral L{f(t)} = … Read more
The Laplace transform of t is equal to L{t} = 1/s2. Here we will find the Laplace of t by the definition of Laplace transforms. The formula of the Laplace of t is given as follows: L{t} = $\dfrac{1}{s^2}$. Find Laplace Transform of t Recall, the definition of the Laplace transform. According to this, the … Read more
The Laplace transform of sin t is equal to 1/(s2+1), and it is denoted by L{sint}. Here we will find the Laplace of sint by the definition of Laplace transforms. The formula of the Laplace of sint is given by L{sint} = $\dfrac{1}{s^2+1}$. Laplace transform of sint By definition, the Laplace transform of f(t) is … Read more
The Laplace transform of 2 is equal to 2/s, and it is denoted by L{2}. Lets find the Laplace of 2 by the definition of Laplace transforms. Recall, the Laplace of f(t) by definition is given by the following integral: L{f(t)} = $\int_0^\infty e^{-st} f(t)\,dt$ Laplace of 2 By the above definition, the Laplace of … Read more
The derivative of sec2x by first principle is equal to 2sec2x tan2x. In this post, we will find the derivative of sec2x by first principle. The first principle rule of derivatives states that the derivative of f(x) is given by the limit f$’$(x) = limh→0 $\dfrac{f(x+h)-f(x)}{h}$. Let us now find the differentiation of sec2x by … Read more
The general solution of the differential equation dy/dx=cos(x+y) is equal to tan(x+y)/2 = x+C. Here we learn how to solve dy/dx=cos(x+y). Find General Solution of dy/dx=cos(x+y) Question: What is the solution of dy/dx=cos(x+y)? Answer: The solution of dy/dx=cos(x+y) is tan(x+y)/2 = x+C. Solution: Let x+y=2u So 1+ $\dfrac{dy}{dx}$ = 2 $\dfrac{du}{dx}$ ⇒ $\dfrac{dy}{dx}$ = 2 … Read more
The integration of sin^3x cos^2x is equal to (cos5x)/5 – (cos3x)/3 +C where C is a constant. Here we learn how to integrate sin^3x cos^2x. What is the Integration of sin3x cos2x? Question: Find the integral of sin3x cos2x, that is, Find ∫sin3x cos2x dx Solution: We have: ∫sin3x cos2x dx = ∫sin2x sin x … Read more
The derivative of x^logx (x to the power logx) is equal to 2 logx xlogx-1. Here we find the differentiation of xlogx by using the product rule of derivatives together with logarithmic differentiation. The product rule of derivatives states the following: $\dfrac{d}{dx}$(f(x)g(x)) = f(x) $\dfrac{d}{dx}$(g(x)) + g(x) $\dfrac{d}{dx}$(f(x)) …(I) Find the Derivative of xlogx Answer: … Read more