Chapter 16

If \([\cdot]\) denotes the greatest integer function, then \(\int_{0}^{2}[x+[x+[x]]] d x=\) (A) 1 (B) 2 (C) 3 (D) 0

\(\lim _{n \rightarrow \infty}\left(\sin \frac{\pi}{2 n} \cdot \sin \frac{2 \pi}{2 n} \cdot \sin \frac{3 \pi}{2 n} \cdots \sin \frac{(n-1) \pi}{n}\right)^{1 / n}\) (A) \(\frac{1}{4}\) (B) 4 (C) 1 (D) None of these

\(\int_{-2 \pi}^{5 \pi} \cot ^{-1}(\tan x) d x\) (A) \(7 \pi^{2}\) (B) \(\frac{7 \pi^{2}}{2}\) (C) 0 (D) \(\frac{3 \pi^{2}}{2}\)

\(\int_{0}^{\sqrt{3}} \frac{1}{1+x^{2}} \cdot \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x\) (A) \(\frac{7}{72} \pi^{2}\) (B) \(\frac{3}{42} \pi^{2}\) (C) \(\frac{17}{72} \pi^{2}\) (D) None of these

The value of the definite integral \(\int_{0}^{1} \frac{x}{x^{2}+16} d x\) lies in the interval \([a, b]\). The smallest such interval is (A) \([0,1]\) (B) \(\left[0, \frac{1}{7}\right]\) (C) \(\left[0, \frac{1}{17}\right]\) (D) None of these

\(\int_{0}^{1} \frac{d x}{\sqrt{4-x^{2}-x^{3}}}\) belongs to the interval (A) \(\left[0, \frac{\pi}{6}\right]\) (B) \(\left[\frac{\pi}{6}, \frac{\pi}{4 \sqrt{2}}\right]\) (C) \(\left[\frac{\pi}{4 \sqrt{2}}, \frac{\pi}{2}\right]\) (D) None of these

The value of a positive integer \(n \leq 5\) such that \(\int_{0}^{1} e^{x}(x-1)^{n} d x=16-6 e\) is (A) 1 (B) 2 (C) 3 (D) 4

Let \(f(x)\) be a function defined by \(f(x)=\) \(\int_{1}^{x} x\left(x^{2}-3 x+2\right) d x, 1 \leq x \leq 3\), then the range of \(f(x)\) is (A) \(\left[-\frac{1}{4}, 2\right]\) (B) \(\left[-\frac{1}{4}, 4\right]\) (C) \([0,2]\) (D) None of these

The value of the integral \(\int_{2}^{3}(\sqrt{2 x-\sqrt{5(4 x-5)}}\) \(+\sqrt{2 x+\sqrt{5(4 x-5)}}) d x\) is equal to (A) \(\frac{7 \sqrt{7}+2 \sqrt{5}}{3 \sqrt{2}}\) (B) \(\frac{7 \sqrt{7}-2 \sqrt{5}}{3 \sqrt{2}}\) (C) \(\frac{2 \sqrt{5}-7 \sqrt{7}}{3 \sqrt{2}}\) (D) None of these

If \(f\) and \(g\) are two continuous functions being even and odd, respectively, then \(\int_{-a}^{a} \frac{f(x)}{b^{g(x)}+1} d x\) is equal to ( \(a\) being any non-zero number and \(b\) is positive real number, \(b \neq 1\) ) (A) Independent of \(f\) (B) Independent of \(g\) (C) Independent of both \(f\) and \(g\) (D) None of these

For \(x>0\), let \(f(x)=\int_{1}^{x} \frac{\ln t}{1+t} d t .\) Then, the value of \(f(e)+f\left(\frac{1}{e}\right)\) is (A) 1 (B) 2 (C) \(\frac{1}{2}\) (D) None of these

The value of the integral \(\int_{0}^{2 \pi} e^{\cos \theta} \cos (\sin \theta) d \theta\) is (A) 0 (B) \(\pi\) (C) \(2 \pi\) (D) cannot be determined

Let \(f\) be a real valued function satisfying \(f(x)+\) \(f(x+6)=f(x+3)+f(x+9) .\) Then, \(\int_{x}^{x+12} f(t) d t\) is (A) A linear function (B) An exponential function (C) A constant function (D) None of these

\(\lim _{n \rightarrow \infty} \frac{\left(1^{2}+2^{2}+3^{2}+\ldots+n^{2}\right)\left(1^{3}+2^{3}+3^{3}+\ldots+n^{3}\right)}{\left(1^{6}+2^{6}+3^{6}+\ldots+n^{6}\right)}\) is equal to (A) \(\frac{7}{12}\) (B) \(\frac{12}{7}\) (C) \(\frac{5}{12}\) (D) None of these

If \(I\) is the greatest of the definite integrals \(I_{1}=\int_{0}^{1} e^{-x} \cos ^{2} x d x, I_{2}=\int_{0}^{1} e^{-x^{2}} \cos ^{2} x d x\) \(I_{3}=\int_{0}^{1} e^{-x^{2}} d x\), and \(I_{4}=\int_{0}^{1} e^{-x^{2} / 2} d x\), then, (A) \(I=I_{1}\) (B) \(I=I_{2}\) (C) \(I=I_{3}\) (D) \(I=I_{4}\)

The values of \(a\) for which the equation \(\int_{0}^{x} \sin ^{2} \frac{t}{2} d t=\) \(a^{2} x^{2}-\frac{1}{2}(3 x-1)+\frac{1}{a^{2}}\) possesses a solution are (A) \(\pm \frac{1}{\sqrt{n \pi+\frac{\pi}{2}}}, n \in N\) (B) \(\pm \frac{1}{\sqrt{2 n \pi+\pi}}, n \in N\) (C) \(\pm \frac{1}{\sqrt{2 n \pi-\frac{\pi}{2}}}, n \in N\) (D) None of these

The value of the integral \(\int_{0}^{\infty} \frac{d x}{\left(x+\sqrt{x^{2}+1}\right)^{n}}\), where \(n>1\), is (A) \(\frac{n^{2}}{n^{2}-1}\) (B) \(\frac{n}{n^{2}-1}\) (C) \(\frac{n^{2}}{n^{2}+1}\) (D) \(\frac{n}{n^{2}+1}\)

If \(\alpha\) is a parameter independent of \(x\) and \(\alpha \neq\) \((2 n+1) \pi, n \in Z\), then the value of the integral \(\int_{0}^{1} \frac{x^{\cos \alpha}-1}{\ln x} d x, x>0, x \neq 1\) is (A) \(\ln (1+\cos \alpha)\) (B) \(\ln (1-\cos \alpha)\) (C) \(\ln |\cos \alpha|\) (D) None of these

The value of \(\int_{0}^{1}(\\{2 x\\}-1)(\\{3 x\\}-1) d x\), where \(\\{\cdot\\}\) denotes the fractional part is, (A) \(\frac{19}{72}\) (B) \(\frac{31}{9}\) (C) \(\frac{1}{8}\) (D) \(\frac{72}{19}\)

The area included between the curves \(x^{2}+y^{2}=a^{2}\) and \(\sqrt{|x|}+\sqrt{|y|}=\sqrt{a}(a>0)\) is (A) \(\left(\pi+\frac{2}{3}\right) a^{2}\) (B) \(\left(\pi-\frac{2}{3}\right) a^{2}\) (C) \(\frac{2}{3} a^{2}\) (D) \(\frac{2 \pi}{3} a^{2}\)

If \(I_{n}=\int_{0}^{\pi / 2} \frac{\sin ^{2} n x}{\sin ^{2} x} d x\), then (A) \(I_{n}=\frac{n \pi}{2}\) (B) \(I_{n}=2 \int_{0}^{\pi / 2} \frac{\sin n x \cos 2 n x}{\sin x} d x\) (C) \(I_{1}, I_{2}, I_{3}, \ldots I_{n}, \ldots\) is an A.P. (D) \(\sin \left(I_{16}\right)=0\)

If \(A_{n}=\int_{0}^{\pi / 2} \frac{\sin (2 n-1) x}{\sin x} d x ;\) \(B_{n}=\int_{0}^{\pi / 2}\left(\frac{\sin n x}{\sin x}\right)^{2} d x ;\) for \(n \in N\), then (A) \(A_{n+1}=A_{n}\) (B) \(B_{n+1}=B_{n}\) (C) \(A_{n+1}-A_{n}=B_{n+1}\) (D) \(B_{n+1}-B_{n}=A_{n+1}\)

If \(g(x)=\int_{0}^{x} \cos 4 t d t\), then \(g(x+\pi)\) equals (A) \(\frac{g(x)}{g(\pi)}\) (B) \(g(x)+g(\pi)\) (C) \(g(x)-g(\pi)\) (D) \(g(x) \cdot g(\pi)\)

If \(x>0\) and \(\int_{0}^{x}[x] d x=[x]\left(\frac{1}{2} A+B\right)\), where \([.]\) denotes the greatest integer function, then (A) \(A=[x]-1\) (B) \(B=x-[x]\) (C) \(A=[x]+1\) (D) \(B=x+[x]\)

If \(\int_{1}^{4}|x-3| d x=2 A+B\), Then (A) \(A=3 / 2, B=4\) (B) \(A=1, B=1 / 2\) (C) \(A=2, B=-3 / 2\) (D) \(A=1 / 2, B=3 / 2\)

If \(\int_{a}^{b}|\sin x| d x=8\) and \(\int_{a}^{a+b}|\cos x| d x=\frac{9}{2}\), then (A) \(a=\frac{\pi}{2}\) (B) \(b=\frac{11 \pi}{4}\) (C) \(a=\frac{\pi}{4}\) (D) \(b=\frac{17 \pi}{4}\)

If \(\int_{0}^{1} \frac{d x}{2 e^{x}-1}=p \log (q e-1)-r\), then (A) \(p=1\) (B) \(q=2\) (C) \(r=1\) (D) \(r=2\)

\(\int_{0}^{\pi / 2} f(\sin 2 x) \sin x d x\) is equal to (A) \(\int_{0}^{\pi / 2} f(\sin 2 x) \cos x d x\) (B) \(\sqrt{2} \int_{0}^{\pi / 4} f(\cos 2 x) \cos x d x\) (C) \(\sqrt{2} \int_{0}^{\pi / 4} f(\cos 2 x) \sin x d x\) (D) None of these

The absolute value of \(\int_{10}^{19} \frac{\sin x}{1+x^{8}} d x\) is (A) less than \(10^{-7}\) (B) more than \(10^{-7}\) (C) less than \(10^{-6}\) (D) more than \(10^{-6}\)

\(\int_{-1 / 2}^{1 / 2} \sqrt{\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2} d x\) is equal to (A) \(4 \log \frac{3}{4}\) (B) \(4 \log \frac{4}{3}\) (C) \(2 \log \frac{16}{9}\) (D) \(-\log \frac{81}{256}\)

If \(I_{n}=\int_{0}^{1} \frac{d x}{\left(1+x^{2}\right)^{n}} ; n \in N\), then (A) \(2 n I_{n+1}=2^{-n}-(2 n-1) I_{n}\) (B) \(2 n I_{n+1}=2^{-n}+(2 n-1) I_{n}\) (C) \(I_{2}=\frac{\pi}{8}+\frac{1}{4}\) (D) \(I_{2}=\frac{\pi}{8}-\frac{1}{4}\)

Given \(f\) is an odd function and periodic with period \(2 .\) If \(f(x)\) is continuous \(\forall x\) and \(g(x)=\int_{0}^{x} f(t) d t\), then (A) \(g\) is an odd function (B) \(g\) is periodic with period 2 (C) \(g(2 n)=0\) (D) \(g(2 n)=1\)

If \(A_{n}\) be the area bounded by the curve \(y=(\tan x)^{n}\) and the lines \(x=0, y=0\) and \(x=\frac{\pi}{4}\), then for \(n>2\), (A) \(A_{n}+A_{n-2}=\frac{1}{n-1}\) (B) \(A_{n}+A_{n+2}=\frac{1}{n+1}\) (C) \(\frac{1}{2 n+1}

If \(f(x)= \begin{cases}\frac{1}{2^{n}}, & \text { when } \frac{1}{2^{n+1}}

If \(f(x)= \begin{cases}\frac{1}{n}, & \frac{1}{n+1}

\(\int_{0}^{1} \frac{\sin \pi x}{1+x^{2}} d x\) lies between (A) \(\frac{1}{\pi}\) and \(\frac{2}{\pi}\) (B) \(\frac{\pi}{2}\) and \(\pi\) (C) \(\pi\) and \(\frac{3 \pi}{2}\) (D) None of these

\(\int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} d x\) lies between (A) \(\frac{\pi^{2}}{3}\) and \(\frac{2 \pi^{2}}{3}\) (B) \(\frac{\pi^{2}}{9}\) and \(\frac{2 \pi^{2}}{9}\) (C) \(\frac{2 \pi^{2}}{9}\) and \(\frac{4 \pi^{2}}{9}\) (D) None of these

\(\int_{0}^{1} \frac{x^{2}}{\sqrt{1+x^{2}}}\) lies between (A) \(\frac{1}{\sqrt{2}}\) and \(\frac{1}{\sqrt{3}}\) (B) \(\frac{1}{\sqrt{2}}\) and 1 (C) \(\frac{1}{3 \sqrt{2}}\) and \(\frac{1}{3}\) (D) None of these

\(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \cdots\left(1+\frac{n}{n}\right)\right]^{1 / n}\) is equal to (A) \(\frac{2}{6}\) (B) \(\frac{e}{2}\) (C) \(\frac{e}{4}\) (D) \(\frac{4}{e}\)

The area enclosed by the curve \(x=a \cos ^{3} t, y=b \sin ^{3}\) \(t, 0 \leq t \leq 2 \pi\) is (A) \(\frac{3 \pi a b}{8}\) (B) \(\frac{3 \pi a b}{4}\) (C) \(\frac{3 a b}{8}\) (D) None of these

If \([\cdot]\) denotes the greatest integer function, then Column-I Column-II I. \(\int_{0}^{\infty}\left[\frac{2}{e^{x}}\right] d x=\) (A) 2 II. \(\int_{0}^{1.5}\left[x^{2}\right] d x=\) (B) \(\ln 2\) III. \(\int_{0}^{\pi / 2} \frac{1+2 \cos x}{(2+\cos x)^{2}} d x=\) (C) \(2-\sqrt{2}\) IV. \(\int_{3}^{4} \frac{\left[x^{2}\right]}{\left[x^{2}-14 x+49\right]+\left[x^{2}\right]} d x=\) (D) \(\frac{1}{2}\) \(([-]\) denotes the greatest integer function)

\(\begin{array}{ll}\text { Column-I } & \text { Column-II }\end{array}\) I. \(\int_{-1}^{3}(|x-2|+[x]) d x=([x]\) stands for (A) 2 greatest integer to \(x\) ) less than or equal II. \(\int_{-1}^{1} \frac{\sin ^{2} x}{\left[\frac{x}{\sqrt{2}}\right]+\frac{1}{2}} d x=([x]\) stands for (B) 3 greatest integer less than or equal to \(x\) ) III. If \(\int_{1}^{b}(b-4 x) d x \geq 6-5 b\), (C) 7 \(b>1\), then \(b=\) IV. If \(I_{1}=\int_{0}^{3 \pi} f\left(\cos ^{2} x\right) d x\) and then \(I_{1}=\) (D) 0 \(I_{2}=\int_{0}^{\pi} f\left(\cos ^{2} x\right) d x\) \(k I_{2}\), where \(k=\)

Assertion: \(I_{n}=\int_{0}^{\infty} x^{n} e^{-x} d x(n\) is a positive integer) \(\begin{aligned} &=n ! \\ \text { Reason: } I_{n}=& n I_{n-1} \end{aligned}\)

Assertion: If \(f(x)\) is a non-negative continuous function such that \(f(x)+f\left(x+\frac{1}{2}\right)=1\), then \(\int_{0}^{2} f(x) d x=1\) Reason: \(f(x)\) is a periodic function having period \(1 .\)

\(\int_{0}^{10 \pi}|\sin x| d x\) is \(\quad[2002]\) (A) 20 (B) 8 (C) 10 (D) 18

\(\int_{0}^{2}\left[x^{2}\right] d x\) is \(\quad[2002]\) (A) \(2-\sqrt{2}\) (B) \(2+\sqrt{2}\) (C) \(\sqrt{2}-1\) (D) \(-\sqrt{2}-\sqrt{3}+5\)

\(\int_{-\pi}^{\pi} \frac{2 x(1+\sin x)}{1+\cos ^{2} x} d x\) is (A) \(\frac{\pi^{2}}{4}\) (B) \(\pi^{2}\) (C) zero (D) \(\frac{\pi}{2}\)

Evaluate \(\int_{0}^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\) (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{2}\) (C) zero (D) 1

The area bounded by the curve \(y=2 x-x^{2}\) and the straight line \(y=-x\) is given by \([2002]\) (A) \(\frac{9}{2}\) sq unit (B) \(\frac{43}{6}\) sq unit (C) \(\frac{35}{6}\) sq unit (D) None of these

If \(f(y)=e^{y}, g(y)=y ; y>0\) and \(F(t)=\int_{0}^{t} f(t-y) g(y) d y\), then \([\mathbf{2 0 0 3}]\) (A) \(F(t)=1-e^{-t}(1+t)\) (B) \(F(t)=e^{t}-(1+t)\) (C) \(F(t)=t e^{t}\) (D) \(F(t)=t e^{-t}\)