Engineering Mat...

If \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {4 - x}&{x \le 2}\\ {kx - 4}&{x > 2} \end{array}} \right.\) is a continuous function for all real values of x, then f(8) is equal to ________.

The integral \(\mathop \smallint \limits_2^\infty \frac{{dx}}{{x\log x}}\)

The value of the following limit is

\(\mathop {\lim }\limits_{z \to 1} \left( {1 - z} \right)\tan \frac{{\pi z}}{2}\)

\(\smallint \frac{{dx}}{{1 + x + {x^2} + {x^3}}}\) is equal to

Evaluate \(\mathop {\lim }\limits_{x \to 0} \frac{{\log x}}{{\cot x}}\;\)

In u = x + 3y² – z³, ν = 4x²yz, w = 2z² – xy evaluate ∂(u, ν, w)/∂(x, y, z) at (1, -1, 0) 

If \(f\left( x \right) = A~\cos\left( {\frac{{2\pi x}}{5}} \right) + C,\;f'\left( {\frac{{15}}{4}} \right) = \frac{1}{2}\) and \(\mathop \smallint \nolimits_0^5 f\left( x \right)dx = \frac{{A\pi }}{2}\) then the value of A and C are respectively.

If \(\mathop \smallint \nolimits_0^{2\pi } \left| {x\;sin\;x} \right|dx = k\pi ,\) then the value of k is equal to ______.

\(\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{x + \sin x}}{x}} \right)\)  equals to

If R is the region 0 ≤ x ≤ y ≤ L, then

\(\mathop \int\!\!\!\int \limits_R \left( {{x^2} + {y^2}} \right)dx\;dy\) is