21. The value of the definite integral $$\int_1^{\text{e}} {\sqrt {\text{x}} } \ln \left( {\text{x}} \right){\text{dx}}$$ is
22. In the Taylor series expansion of ex about x = 2, the coefficient of (x - 2)4 is
23. The value of the integral $$\int_0^\pi {{\text{x}}{{\cos }^2}{\text{xdx}}} $$ is
24. The value of the line integral $$\int\limits_{\text{c}} {\left( {2{\text{x}}{{\text{y}}^2}{\text{dx}} + 2{{\text{x}}^2}{\text{ydy}} + {\text{dz}}} \right)} $$ along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is
25. Consider a vector field $$\overrightarrow {\text{A}} \left( {\overrightarrow {\text{r}} } \right).$$ The closed loop line integral $$\oint {\overrightarrow {\text{A}} \cdot \overrightarrow {{\text{d}}l} } $$ can be expressed as
26. The area between the parabolas y2 = 4ax and x2 = 4ay is
27. $$\mathop {\lim }\limits_{{\text{x}} \to \infty } {{\text{x}}^{\frac{1}{{\text{x}}}}}$$ is
28. For a position vector \[{\rm{r}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\] the norm of the vector can be defined as $$\left| {\overrightarrow {\text{r}} } \right| = \sqrt {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} .$$ Given a function $$\phi = \ln \left| {\overrightarrow {\text{r}} } \right|,$$ its gradient $$\nabla \phi $$ is
29. According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value $$\xi $$ in this interval such that $$\int\limits_{\text{a}}^{\text{b}} {{\text{f}}\left( {\text{x}} \right){\text{dx}} = } $$
30. Let $$\nabla \cdot \left( {{\text{f}}\overrightarrow {\text{v}} } \right) = {{\text{x}}^2}{\text{y}} + {{\text{y}}^2}{\text{z}} + {{\text{z}}^2}{\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\overrightarrow {\text{v}} = {\text{y}}\overrightarrow {\text{i}} + {\text{z}}\overrightarrow {\text{j}} + {\text{x}}\overrightarrow {\text{k}} ,$$ then $$\overrightarrow {\text{v}} \cdot \nabla {\text{f}}$$ is
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