The value of \int\limits_0^{\frac{\pi }{6}} {\sin x\cos x\,dx} \,\_\_\_\_\_\_\_.

• A) \frac{1}{8}
• B) \frac{1}{4}
• C) 4
• D) 8

If the average value of y = sin3x with respect to x over [0, 2] is 0.525, then what will be the value of $\int\limits_0^2 {{{\sin }^3}xdx} $?

• A) 2.5
• B) 1.05
• C) 1.5
• D) 0.5

What could be the value of x if \int\limits_0^x {3dx} > 15 ?

• A) x>5
• B) x>15
• C) x>3
• D) x>10

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

• A) 1
• B) 2
• C) 0
• D) 3

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

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

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) \[\int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = [\cos x]_0^1 + [\tan x]_0^1\]

$\int\limits_a^b {f(x)dx = \_\_\_\_\_\_.} $

• A) $ - \int\limits_b^a {f(x)dx} $
• B) $\int\limits_b^a {f(x)dx} $
• C)
• D)

In integration of f(x)=x{{({{x}^{2}}-3)}^{4}} from x=0 to x=2 by substitution method, we take u={{x}^{2}}-3 then du= ................

• A) dx
• B) x
• C) 2xdx
• D) 2x

The value of $\int\limits_1^x {{y^2}dy = \_\_\_\_\_.} $

• A) $\frac{{{y^3}}}{3} - \frac{1}{3}$
• B) $\frac{{{x^3}}}{3} - \frac{1}{3}$
• C)
• D)

If \[ \int_0^2 {(x^2 + 1)} dx = \frac{{14}} {3} \] Then the solution of \[ \int_2^0 {(x^2 + 1)} dx = \] will be....

• A) -14/3
• B) 14/3
• C) -3/14
• D) none of these

\[ \begin{gathered} {\text{If the upper and lower limits for the definite integral are the same, then }} \ \int_a^a {f(x)} dx = \\ \end{gathered} \]

• A) none of these
• B) negative integer
• C) zero
• D) positive integer

If “f” is a continuous function on [a,b] then \[\int\limits_a^b {f(x)\,dx = } \,\_\_\_\_\_\_.\]

• A) \[\int\limits_{b}^{b}{f(x)dx}\]
• B) \[ - \int\limits_a^b {f(x)\,dx} \]
• C) \[ - \int\limits_c^a {f(x)\,dx} \]
• D) \[ - \int\limits_b^a {f(x)\,dx} \]

For the adjacent intervals, [a,c] and [c,b],where c is any number,\[ \int_a^b {f(x)} dx = \]

• A) \[ \int_a^c {f(x)} dx + \int_c^b {f(x)} dx \]
• B) None of these
• C) \[ \int_a^b {f(x)} dx + \int_c^a {f(x)} dx \]
• D) \[ \int_a^c {f(x)} dx + \int_b^a {f(x)} dx \]

What will be the average value of y = cos3x with respect to x over [0, 2], if $\int\limits_0^2 {{{\cos }^3}xdx} $ is equal to 0.66?

• A) 1.32
• B) 1.05
• C) 1.5
• D) 0.33

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

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

The value of \int\limits_0^1 {\frac{{dx}}{{1 + {x^2}}}} \,\_\_\_\_\_.

• A) \frac{\pi }{2}
• B) \infty
• C) \frac{\pi }{4}
• D) 0

The value of\[\int\limits_{ - 2}^2 {|x|\,dx} \,\_\_\_\_\_\_.\]

• A) None of the above
• B) 4
• C) 0
• D) 2

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

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

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

• A) 2/3
• B) 1/4
• C) 1/3
• D) 4/3

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

• A) \[\int\limits_0^1 {(\sin x\cos x)dx} = 2\int\limits_0^1 {\sin xdx} \times 2\int\limits_0^1 {\cos xdx} \]
• B) \[\int\limits_0^1 {(\sin x\cos x)dx} = \int\limits_0^1 {\sin xdx} + \int\limits_0^1 {\cos xdx} \]
• C) \[\int\limits_0^1 {(\sin x\cos x)dx} = \frac{1}{2}\int\limits_0^1 {\sin 2xdx} \]
• D) \[\int\limits_0^1 {(\sin x\cos x)dx} = \int\limits_0^1 {\sin xdx} - \int\limits_0^1 {\cos xdx} \]