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]

{\text{The integral }}\int {{{\left( {{x^3} + 1} \right)}^{10}}\,.3{x^2}\,dx\,} {\text{will be equal to ?}}

• A) - \frac{{{{\left( {{x^3} + 1} \right)}^{11}}}}{{11}} + c
• B) {\text{None of these}}
• C) \frac{{{{\left( {{x^3} + 1} \right)}^{11}}}}{{11}} + c
• D) \frac{{{{\left( {{x^3} - 1} \right)}^{11}}}}{{11}} + c

{\text{The integral }}\int {\frac{{6x}}{{{{(3{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}

• A) - \frac{1}{{6x + 1}} + c
• B) \frac{1}{{3{x^2} + 1}} + c
• C) - \frac{1}{{{{(3{x^2} + 1)}^3}}} + c
• D) - \frac{1}{{3{x^2} + 1}} + c

{\text{The integral }}\int {\frac{{4x}}{{{{(2{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}

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

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

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

{\text{The}}\,{\text{value}}\,{\text{of}}\,c\,{\text{in}}\,{\text{Rolle's}}\,{\text{theorem}}\,{\text{for}}\,{\text{the}}\,{\text{function}}\,f(x) = {x^3} - 3x\,{\text{in}}\,{\text{the}}\,{\text{interval}}\,\left[ {0,\sqrt 3 } \right]{\text{.}}

• A) - 1
• B) 1
• C) \frac{2}{3}
• D) \frac{1}{3}

Evaluate\;\int_0^x {\cos t} dt =

• A) {\cos x}
• B) \sin t
• C) \sin x
• D) {\cos t}

The value of $\int\limits_1^2 {dx = \_\_\_\_\_\_.} $

• A) 1
• B) 2
• C) 0
• D) 3

If the function and limits of definite integral are the same and variable of integration are changed, i.e \int_a^b {f(x)} dx = \int_a^b {f(t)} dt Then the answer would be

• A) changed
• B) do not changed
• C)
• D)

Which of the following statements is true about \int\limits_0^1 {(\cos x + {{\sec }^2}x)dx}?

• A) \int\limits_0^1 {(\cos x + {{\sec }^2}x)dx} = [\sin x]_0^1 + [\tan x]_0^1
• B) \int\limits_0^1 {(\cos x + {{\sec }^2}x)dx} = [\sin x]_0^1 \times [\tan x]_0^1
• C) \int\limits_0^1 {(\cos x + {{\sec }^2}x)dx} = [\sin x]_0^1 - [\tan x]_0^1
• D) None

\begin{gathered} If\;\;f(x) \geqslant g(x)\;for\;any\;two\;number\;such\;that\;,a \leqslant x \leqslant b,\;we\;have, \ \int_a^b {f(x)} dx \\ \end{gathered}

• A) \geqslant \int_a^b {g(x)} dx
• B) none of these
• C) \leqslant \int_a^b {g(x)} dx
• D) = \int_a^b {g(x)} dx

\[ If\;\;m \leqslant f(x) \leqslant M\;for\;any\;two\;number\;such\;that\;,a \leqslant x \leqslant b,\;which\;of\;the\;following\;is\;true \]

• A) \[ m(b - a) \geqslant \int_a^b {f(x)} dx \leqslant M(b - a) \]
• B) \[ m(b - a) \leqslant \int_a^b {f(x)} dx \leqslant M(b - a) \]
• C) none of these
• D) \[ m(b - a) \geqslant \int_a^b {f(x)} dx \geqslant M(b - a) \]

\[ Evaluate\;\int_0^x {\sin t} dt = \]

• A) \[ 1 + \cos t \]
• B) \[ 1 + \cos t \]
• C) \[ 1 - \cos x \]
• D) \[ 1 + \cos x \]

Which of the following statements is true about \int\limits_0^1 {(\sin x - {{\sec }^2}x)dx}?

• A) \int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = [\cos x]_0^1 - [\tan x]_0^1
• B) None
• C) \int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = [\cos x]_0^1 + [\tan x]_0^1
• D) \int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = - \,[\cos x]_0^1 - [\tan x]_0^1