Let ∫ dx (√x - √x-1 )^2 = α u(x) + β v(x) + C, where u(x) = x^2 - x + (x - 1 2 )√x^2 - x and v(x) = log |x - 1 2 + √x^2 - x |. The value of α + β is :

The derivative of x³ + 2x² - 5x + 1 is:

If the function $f(x) = 2x^3 + 9x^2 + 12x-1$ is given,then $f(x)$ have

If $\int \frac{dx}{(x-1)^3/^4. (x+2)^5/^4} = a[1 - g(x)]^b + c$, where $c$ is a constant of integration, then which of the following are true? (A) $a = \frac{2}{3}$ (B) $\beta = \frac{3}{4}$ (C) $3\alpha + 4\beta = 5$ (D) $g(x) = \frac{3}{(x+2)}$ Choose the **correct** answer from the options given below:

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Function** | **Increasing on the interval** | | (A) $f(x) = -x^2 - 2x + 1$ | (I) $(-\infty, -1)$ | | (B) $f(x) = x^2 + 1$ | (II) $(1, \infty)$ | | (C) $f(x) = x^2 - 2x + 3$ | (III) $(-\infty, 0)$ | | (D) $f(x) = -x^2$ | (IV) $(0, \infty)$ | Choose the correct answer from the options given below:

The integral $\int e^x\left(\frac{x-1}{2x^2}\right)dx$ is equal to