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)

Evaluate\;\frac{d} {{dx}}\int_2^x t dt

• A) x^3
• B) x^2
• C) x
• D) none of these

The value of \int\limits_1^2 {\ln x\,dx} \,\_\_\_\_.

• A) 2ln2+2
• B) 2ln2-1
• C) ln2-1
• D) 2ln2+1

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) 1.5
• B) 0.5
• C) 2.5
• D) 1.05

Which of the following statements is true about [\sin x]_0^2 - [\tan x]_0^2?

• A) [\sin x]_0^2 - [\tan x]_0^2 = [2\sin x - 2\tan x]_0^2
• B) [\sin x]_0^2 - [\tan x]_0^2 = [\sin x - \tan x]_0^2
• C) [\sin x]_0^2 - [\tan x]_0^2 = [2\sin x + 2\tan x]_0^2
• D) None

What will be the value of \int\limits_0^2 {(\sin x + 3)dx} if \int\limits_0^2 {(10\sin x + 30)dx = 74} ?

• A) 7.4
• B) 4.2
• C) 5.4
• D) 3.6

\frac{{{x^4}}}{4} - \frac{1}{4} = \_\_\_\_\_.

• A) \int\limits_1^x {{t^3}dt}
• B) \int\limits_1^x {{t^4}dt}
• C)
• D)

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

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

The value of \int\limits_0^1 {{e^{ - x}}\,dx} \,\_\_\_\_\_\_.

• A) \frac{{1 - e}}{e}
• B) \frac{{1 + e}}{e}
• C) \frac{{e - 1}}{e}
• D) None of the above

Which of the following statements is true about \int\limits_0^1 {\sec x\tan xdx}?

• A) \int\limits_0^1 {\sec x\tan xdx} = [\sec x]_0^1 \times [\tan x]_0^1
• B) \int\limits_0^1 {\sec x\tan xdx} = [\sec x]_0^1 + [\tan x]_0^1
• C) \int\limits_0^1 {\sec x\tan xdx} = [\sec x]_0^1
• D) None

We can break up definite integrals across a sum or difference \int_a^b {f(x) \pm g(x)} dx = as

• A) \int_a^b {f(x)dx \pm \int_a^b {g(x)} } dx
• B) \int_b^a {f(x)dx \pm \int_a^b {g(x)} } dx
• C) \int_b^a {f(x)dx \pm \int_b^a {g(x)} } dx
• D) None of these

We can break up definite integrals across a sum or difference \[ \int_a^b {f(x) \pm g(x)} dx = \] as

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

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

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

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

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