\(\int \frac{xe^x}{(x+1)^2}\,dx =\)

The area enclosed by the curve \(y^2 = 4ax\) and its latus rectum is

The solution of the differential equation \((x + 1)\frac{dy}{dx} = 1 + y\) is

Match List-I with List-II List-I | List-II (A) If \(x = t^2\) and \(y = t^3\) then \(\frac{d^2y}{dx^2}\) at \(t = 1\) | (i) -2 (B) If \(f(x) = \sqrt{x} + 1\), the \(f^{\prime}(1)\) | (ii) -1 (C) The minimum value of \(f(x) = 9x^2 + 12x + 2\) is | (iii) \(\frac{3}{4}\) (D) The poi…

Order and degree of the differential equation \(y\frac{dy}{dx} + \frac{4}{\frac{dy}{dx}} = 5\) are

The given function \(f(x) = x^5 - 5x^4 + 5x^3 - 1\) has/have (A) Local maxima at \(x = 1\) (B) Local maximum value is 0 (C) Local minimum at \(x = 3\) (D) Local minimum value is -28 (E) The point of inflexion is \(x = 1\) Choose the correct answer from the options given below

Derivative of \(x^3 + 1\) with respect to \(x^2 + 1\) is

Consider the function \(f(x) = x^{\frac{1}{x}}\). Its

Solution of the differential equation \((x + xy)dy - y(1 - x^2)dx = 0\) is