$${\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{differentiable}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}$$

• A) $$(a,b]$$
• B) $$[a,b)$$
• C) $$(a,b)$$
• D) $$[a,b]$$

{\text{What}}\,{\text{does}}\,{\text{the}}\,{\text{indefinite}}\,{\text{integral }}\int_{}^{} {f(x)} dx\,{\text{represent?}}

• A) {\text{None}}\,{\text{of}}\,{\text{these}}
• B) {\text{Area}}\,{\text{under}}\,{\text{the}}\,{\text{curve}}
• C) {\text{Families}}\,{\text{of}}\,{\text{antiderivative}}\,{\text{of}}\,{\text{the}}\,{\text{function }}f(x)
• D) {\text{Curvature}}\,{\text{of}}\,{\text{the}}\,{\text{curve}}

{\text{The integral }}\int {\sqrt {4x - 3} \,dx\,} {\text{will be equal to ?}}

• A) {\text{None of these}}
• B) \frac{{{{(4x + 3)}^{\frac{3}{2}}}}}{6} + c
• C) \frac{{{{(4x - 3)}^{\frac{3}{2}}}}}{3} + c
• D) \frac{{{{(4x - 3)}^{\frac{3}{2}}}}}{6} + c

{\text{The integral }}\int {\cos (5x)\,dx\,} {\text{will be equal to ?}}

• A) - \frac{{\sin (5x)}}{5} + c
• B) {\text{None of these}}
• C) \frac{{\sin (5x)}}{5} + c
• D) 5\sin (5x) + c

{\text{The integral }}\int {\frac{x}{{1 + {x^2}}}\,dx\,} {\text{will be equal to ?}}

• A) 2\ln (1 + {x^2}) + c
• B) \frac{{\ln \left( {1 - {x^2}} \right)}}{2} + c
• C) {\text{ln(1 + }}{{\text{x}}^2}) + c
• D) \frac{{\ln \left( {1 + {x^2}} \right)}}{2} + c

{\text{The integral }}\int {\cot (2x)\,dx\,} {\text{will be equal to ?}}

• A) \frac{1}{2}\ln \left| {\sin (2x)} \right| + c
• B) \frac{1}{2}\ln \left| {\sec (2x)} \right| + c
• C) {\text{ln}}\left| {\sin (2x)} \right| + c
• D) \ln \left| {{\text{sec}}(2x)} \right| + c

The Area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:

• A) A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \left[ {f\left( {{x_1}} \right)\Delta x + f\left( {{x_2}} \right)\Delta x + \ldots + f\left( {{x_{n - 1}}} \right)\Delta x} \right]
• B) A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Delta x}}\;\left[ {f\left( {{x_1}} \right) + f\left( {{x_2}} \right) + \ldots + f\left( {{x_n}} \right)} \right]
• C) A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \Delta x\left[ {f\left( {{x_1}} \right) + f\left( {{x_2}} \right) + \ldots + f\left( {{x_n}} \right)} \right]
• D) A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \left[ {f\left( {{x_1}} \right) + f\left( {{x_2}} \right) + \ldots + f\left( {{x_n}} \right)} \right]

{\text{The integral }}\int {\frac{{10x}}{{{{(5{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}

• A) - \frac{1}{{5{x^2} + 1}} + c
• B) - \frac{1}{{5x + 1}} + c
• C) \frac{1}{{5{x^2} + 1}} + c
• D) - \frac{1}{{{{(5{x^2} + 1)}^3}}} + c

{\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{differentiable}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}

• A) (a,b]
• B) [a,b]
• C) (a,b)
• D) [a,b)

\int {\tan x} dx = \_\_\_\_\_\_\_\_\_\_\_\_\_\_.

• A) \ln \left| {\sin x} \right| + C
• B) \ln \left| {\sec x} \right| + C
• C) \ln |\cos x| + C
• D) {\sec ^2}x + C

{\text{The integral }}\int {{{\left( {{x^2} + 1} \right)}^{\frac{5}{2}}}.\,x\,dx\,} {\text{will be equal to ?}}

• A) {\left( {{x^2} + 1} \right)^{\frac{7}{2}}} + c
• B) \frac{{2{{\left( {{x^2} + 1} \right)}^{\frac{7}{2}}}}}{7} + c
• C) {\text{None of these}}
• D) \frac{{{{\left( {{x^2} + 1} \right)}^{\frac{7}{2}}}}}{7} + c

{\text{The integral }}\int {\cos e{c^2}(3{x^2})\,.6x\,dx\,} {\text{will be equal to ?}}

• A) \sec (3{x^2}) + c
• B) - \cot (3{x^2}) + c
• C) {\text{None of these}}
• D) {\text{cot(3}}{{\text{x}}^2}) + c,

{\text{The integral }}\int {\sqrt {2x + 3} \,dx} {\text{ will be equal to ?}}

• A) \frac{{{{\left( {2x + 3} \right)}^{\frac{1}{2}}}}}{3} + c
• B) \frac{{{{\left( {2x + 3} \right)}^{\frac{3}{2}}}}}{3} + c
• C) \frac{{{{\left( {2x + 3} \right)}^{\frac{3}{2}}}}}{2} + c
• D) \frac{{{{\left( {2x + 3} \right)}^{\frac{2}{3}}}}}{3} + c

The symbol \int {} was introduced by___________and is called integral sign.

• A) Newton
• B) Lagrange
• C) Cauchy
• D) Leibnitz

{\text{The}}\,{\text{value}}\,{\text{of}}\,c\,{\text{in}}\,{\text{Rolle's}}\,{\text{Theorem}}\,{\text{for}}\,{\text{the}}\,{\text{function}}\,f(x) = {e^x}\sin x,\,\,x \in [0,\pi ]\,{\text{is}}\_\_\_\_\_\_\_\_\_\_.

• A) \frac{\pi }{6}
• B) \frac{{3\pi }}{2}
• C) \frac{{3\pi }}{4}
• D) \frac{\pi }{4}

{\text{In}}\,{\text{the}}\,{\text{notation:}}\,\int_a^b {f(x)} dx{\text{,}}\;f(x)\,{\text{is}}\,{\text{called___________}}{\text{.}}

• A) {\text{Differential}}
• B) {\text{None}}\,{\text{of}}\,{\text{these}}
• C) {\text{Integration}}
• D) {\text{Integrand}}

{\text{The integral }}\int {{{\sec }^2}(5{x^2})\,.10x\,dx\,} {\text{will be equal to ?}}

• A) {\text{se}}{{\text{c}}^2}(10x) + c
• B) \tan (5{x^2}) + c
• C) \sec (5{x^2}).\tan (5{x^2}) + c
• D) \tan (5x) + c

{\text{The integral }}\int {{{\sec }^2}(2{x^2})\,.4x\,dx\,} {\text{will be equal to ?}}

• A) \tan (2x) + c
• B) {\text{se}}{{\text{c}}^2}(2x) + c
• C) \tan (2{x^2}) + c
• D) \sec (2{x^2}).\tan (2{x^2}) + c

{\text{The integral }}\int {\sin (5x)dx} {\text{ will be equal to ?}}

• A) \frac{{\cos 5x}}{5} + c
• B) {\text{ - }}\frac{{\cos 4x}}{5} + c
• C) {\text{ - }}\frac{{\cos 5x}}{5} + c
• D) 5\cos 5x + c

{\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{continuous}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}

• A) [a,b)
• B) (a,b]
• C) (a,b)
• D) [a,b]