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

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

\[{\text{The integral }}\int {\sec (x).\tan (x)\,dx\,} {\text{will be equal to ?}}\]

• A) \[\sec (x) + c\]
• B) \[ - \ln \left| {\cos (x)} \right| + c\]
• C) \[\cos ec(x) + c\]
• D) \[{\text{None of these}}\]

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

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

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

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

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

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

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

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

If the closed interval [-2,2] is divided into ‘50’ equally spaced sub-intervals then the width of each sub-interval is ------------

• A) -4/25
• B) 2/25
• C) 1/25
• D) 4/25

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

• 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