{\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

What does the definite integral \int_a^b {f(x)} dx represent?

• A) The maximum value of the function f(x) from x = a to x = b
• B) The derivative of the function f(x) at the point x = a
• C) The slope of the curve f(x) from x = a to x = b
• D) The area under the curve f(x) from x = a to x = b

If\;f\;continuous\;on\;[a,b]\;and\;F(x) = \int_a^x {f(t)dt} ,\;then\;

• A) none of these
• B) F^/ (x) = f(t)\;on\;\;[a,b]
• C) F^/ (t) = f(x)\;on\;\;[a,b]
• D) F^/ (x) = f(x)\;on\;\;[a,b]

What will be the value of \int\limits_0^1 {{e^x}dx} ?

• A) ex
• B) ec+1
• C) e
• D) 1

The value of the \int\limits_0^1 {{t^3}dt = \_\_\_\_\_\_\_.}

• A) 2/3
• B) 1/4
• C) 4/3
• D) 1/3

The value of \int\limits_0^x {{t^3}dt = \_\_\_\_\_\_.}

• A) \frac{{{x^4}}}{4}
• B) \frac{{{t^4}}}{4}
• C) \frac{{{t^4}}}{4} - 1
• D) None of the above

\[{\text{If }}f{\text{ is continuous at every point of }}[a,b]{\text{ and }}F{\text{ is anti - derivative of }}f{\text{ on }}[a,b]{\text{, then}}\]

• A) \[ \int_a^b {f(x)} dx = F(b) - F(a) \]
• B) \[ \int_a^b {f(x)} dx = F(a) + F(b) \]
• C) \[ \int_a^b {f(x)} dx = F(a) - F(b) \]
• D) none of these

For any constant number c,\int_a^b {cf(x)} dx = which of the following is correct

• A) c\int_a^b {f(x)} dx
• B) \int_a^b {f(x)} dx + c
• C)
• D)