Evaluate the integral: \[ I = \int \frac{\cos x + x \sin x}{x (x + \cos x)} dx =\]
Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \]
Evaluate the integral: \[ \int \frac{2x^2 - 3}{(x^2 - 4)(x^2 + 1)} \,dx = A \tan^{-1} x + B \log(x - 2) + C \log(x + 2) \] Given that, \[ 64A + 7B - 5C = ? \]
If \( x, y \) are two positive integers such that \( x + y = 20 \) and the maximum value of \( x^3 y \) is \( k \) at \( x = a, y = \beta \), then \( \frac{k}{\alpha^2 \beta^2} = ? \)
If the function f(x) = \(\sqrt{x^2 - 4}\) satisfies the Lagrange’s Mean Value Theorem on \([2, 4]\), then the value of \( C \) is}
The angle between the curves \( y^2 = 2x \) and \( x^2 + y^2 = 8 \) is
If \( T = 2\pi \sqrt{\frac{L}{g}} \), \( g \) is a constant and the relative error in \( T \) is \( k \) times to the percentage error in \( L \), then \( \frac{1}{k} = \) ?
If \( y = x - x^2 \), then the rate of change of \( y^2 \) with respect to \( x^2 \) at \( x = 2 \) is:
For \( x<0 \), \( \frac{d}{dx} [|x|^x] \) is given by:
If \( y = \tan(\log x) \), then \( \frac{d^2y}{dx^2} \) is given by: