promath 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
The derivative of cos cube x (cos^3x) is equal to -3cos2x sinx. In this post, we will find the derivative of cos cube x using the chain rule. The derivative formula of cos3x is given as follows: d/dx (cos3x) = -3cos2x sinx. Derivative of Cos^3x The derivative of cos3x is -3cos2x sinx. Explanation: As cos3x … Read more
The cos^3x formula is given by cos3x = (cos3x + 3cosx)/4. In this post, we learn how to prove the cos cube x formula. What is cos cube x formula Answer: cos3x=(cos3x + 3cosx)/4 is the formula of cos cube x. Proof: We will follow the below steps in order to prove cos cube x … Read more
The sin cube x formula is given as follows: sin3x = (3sinx -sin3x)/4, that is, sin^3x formula is equal to (3sinx – sin3x) divided by 4. Here we establish the formula of sine cube x and give a proof. What is sin cube x formula Answer: sin3x = (3sinx -sin3x)/4 Proof: To establish the formula … Read more
Sin3x formula in terms of cosx is expressed as follows: sin3x = (4cos2x-1)√(1-cos2x). Here we learn how to prove sin3x formula in terms of cosine of x. sin3x in Terms of cos x In order to express sin3x formula in terms of cosx, we will use the following trigonometric formulae: Using the above formulas, we … Read more
The formula of cos^2x – sin^2x is equal to cos2x. Let us learn how to establish the proof the identity of cos2x-sin2x. cos2x-sin2x formula is given by cos2x-sin2x = cos2x. Proof of cos2x-sin2x Formula Question: Prove that cos2x-sin2x = cos2x. Proof: cos2x-sin2x = cosx ⋅ cosx – sinx ⋅ sinx = cos(x+x), here we have … Read more