promath Answer: 6/8 as a fraction simplified in its reduced form is equal to 3/4. Here, we learn how to simplify 6/8 into its simplest form. What is 6/8 in Simplest Form? Answer: 6/8 in simplest form is equal to 3/4. Solution: Method 1: Division Method To simplify the fraction 6/8, let us use the division … Read more
Fractions with the same denominator are called like fractions. For example, 1/5 and 4/5. On the other hand, fractions with different denominators are called unlike fractions. For example, 1/5 and 4/7. Here, let us learn like fractions and unlike fractions with definition, examples and their differences. Definition of Like Fractions Two fractions are said to … Read more
Answer: The limit of x^x as x approaches 0+ is equal to 1. That is, limx→0+ xx = 1. Limit of xx when x tends to 0+ Question: Find the limit of xx when x tends to 0+? Solution We need to find limx→0+ xx. Step 1: Note that x = eln x ⇒ $x^x … Read more
Answer: The derivative of x^2 sin(1/x) at x=0 is equal to 0 if the function is defined to be 0 at x=0. Question Find the derivative of the function f(x) = $\begin{cases} x^2 \sin(\frac{1}{x}) & \text{if } x \neq 0 \\ 0 & \text{if } x=0 \end{cases}$ at x=0. That is, find f'(0). Solution From … Read more
The integral of e^(-x^2) from negative infinity to infinity is equal to √π, that is, $\int_{-\infty}^\infty$exp(-x^2) dx = √π. Here we will learn how to integrate e-x^2 (e to the power minus x2) from -∞ to ∞. Integral of exp(-x^2) from Minus Infinity to Infinity Answer: The integral of $e^{-x^2}$ from -∞ to ∞ is … Read more
The integral of e^(-x^2) from 0 to infinity is equal to √π/2, that is, ∫exp(-x^2) dx = √π/2. Here we will learn how to integrate e-x^2 (e to the power -x2) from 0 to ∞. Integration of exp(-x^2) from 0 to infinity Answer: The integral of $e^{-x^2}$ from 0 to infinity is equal to √π/2. … Read more
The Laplace transform of sint/t is equal to tan-1(1/s), that is, L{sint/t} = tan-1(1/s). The formulas for the Laplace of sint/t and the Laplace of sin(at)/t are given as follows: $\boxed{\mathcal{L}\left\{\dfrac{\sin t}{t} \right\}=\tan^{-1}\left(\dfrac{1}{s} \right)}$ and $\boxed{\mathcal{L}\left\{\dfrac{\sin at}{t} \right\}=\tan^{-1}\left(\dfrac{a}{s} \right)}$ Lets learn how to find Laplace of sin(t)/t. Laplace of sint/t Question: What is the Laplace … Read more
In this page, we solve dy/dx+y/x=x^2y^6. This is a Bernoulli’s differential equation. The general solution of dy/dx+y/x=x^2y^6 is 5x3y5 + Cx5y5 = 2 where C denotes an arbitrary constant of integrals. Let us now solve the Bernoulli’s differential equation dy/dx+y/x=x2y6. Solution of dy/dx+y/x=x^2y^6 Question: Find the solution of $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$. Answer: $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$ …(I) The given differential … Read more
The general solution of (D2+4)y=cos2x is given by y=c1cos2x + c2sin2x + (1+xsin2x)/8. Lets learn how to solve (D^2+4)y=cos^2x. As D2 = d2/dx2, the differential equation (D2+4)y=cos2x can also be written as follows: d2y/dx2 + 4y = cos2x. Solve (D2+4)y=cos^2x Question: Solve the equation $\dfrac{d^2y}{dx^2} + 4y = \cos^2x$. Solution: Letting D2 = d2/dx2, we … Read more
Here we learn how to solve (D^2+4)y=sin^2x. The general solution of (D2+4)y=sin2x is given by y=c1cos2x + c2sin2x + (1-xsin2x)/8. The differential equation (D2+4)y=sin2x can also be written as follows: d2y/dx2 + 4y = sin2x. Now let us solve this equation. Solve (D2+4)y=sin^2x Question: Solve the equation $\dfrac{d^2y}{dx^2} + 4y = \sin^2x$. Solution: One can … Read more