Quizzes : Definite Integral

QUIZ-I

Q:1. A function is defined as $\displaystyle f(x) = \left\{\begin{array}{ll} 1 \; , x \geq 0 \\ -1 \; , x < 0 \end{array} \right. $ ; then the value of $\displaystyle \int_{-1}^{1} \frac{[x]}{\frac{x-f(x) + n-1}{n} + \frac{x-f(x) + n}{n} + \frac{x-f(x) + n+11}{n} + \frac{x-f(x) + n+2}{n}}$ is

(A) –1/3

(B) 1/4

(C) 1

(D) none of these

Click to See Answer :
Ans: (A)

 

Q:2. If $\displaystyle \int_{-\pi/4}^{3\pi/4} \frac{e^{\pi/4} dx }{(e^x + e^{\pi/4}) (sin x + cos x) } = k \int_{-\pi/2}^{\pi/2} sec x dx $ ;then the value of k is

(A) $\frac{1}{2}$

(B) $\frac{1}{\sqrt{2}}$

(C) $\frac{1}{2 \sqrt{2}}$

(D) $ – \frac{1}{\sqrt{2}}$

Click to See Answer :
Ans: (C)

 

Q:3. Suppose g(x) satisfies $\displaystyle g(x) = x + \int_{0}^{1} (x y^2 + y x^2 ) g(y) dy $ ; the g(x) is

(A) $ x + \frac{61}{119} x + \frac{80}{119} x^2 $

(B) $ x + \frac{7}{11} x^2 $

(C) $ \frac{180}{61} x + \frac{80}{119} x^2 $

(D) none of these

Click to See Answer :
Ans: (A)

 

Q:4. The value of $\displaystyle \int_{-1}^{0} \frac{x^2 + 2 x}{ln(x+1)} dx $ is equal to

(A) ln2

(B) ln4

(C) ln3

(D) ln6

Click to See Answer :
Ans: (C)

 

Q:5. Let $\displaystyle f(x) = \int_{1}^{x} x(x^2 – 3 x + 2) dx , 1 \le x \le 4 $ . Then the range of f(x) is

(A) [-1/4 , 63/4]

(B) [-1/4 , 2]

(C) [-1/4 , 63/2]

(D) None of these

Click to See Answer :
Ans: (A)

 

Q:6. The value of $\displaystyle \int_{0}^{1} tan^{-1}(\frac{2x -1}{1 + x – x^2}) dx $ is equal to

(A) 1

(B) 0

(C) –1

(D) none of these

Click to See Answer :
Ans: (B)

 

Q:7. $\displaystyle \int_{\pi/4}^{\pi/3} \frac{\phi d\phi}{1 + sin\phi}$ is equal to

(A) $\pi (\sqrt{2} + 1) $

(B) $\pi (\sqrt{2} – 1) $

(C) $\pi (1 – \sqrt{2} ) $

(D) $\frac{\pi}{2} (\sqrt{2} + 1) $

Click to See Answer :
Ans: (B)