Chapter 16

\(\int_{1}^{4}(\\{x\\})^{[x]} d x\), where \(\\{\cdot\\}\) and \([\cdot]\) denote the fractional part and greatest integer function, respectively, is equal to (A) 1 (B) \(\frac{12}{13}\) (C) \(\frac{13}{12}\) (D) \(\frac{6}{7}\)

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 intercepts on \(x\)-axis made by tangents to the curve, \(y=\int_{0}^{x}|t| d t, x \in R\), which are parallel to the line \(y=2 x\), are equal to (A) \(\pm 2\) (B) \(\pm 3\) (C) \(\pm 4\) (D) \(\pm 1\)

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 (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

Let \(f(x)\) be positive, continuous and differentiable on the interval \((a, b)\) and \(\lim _{x \rightarrow a^{+}} f(x)=1, \lim _{x \rightarrow b^{-}} f(x)\) \(=3^{1 /} 4 .\) If \(f^{\prime}(x) \geq f^{3}(x)+\frac{1}{f(x)}\), then the greatest value of \(b-a\) is (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{6}\) (C) \(\frac{\pi}{24}\) (D) \(\frac{\pi}{12}\)

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) \(r=2\)

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) \(g(2 n)=1\)

A function is said to be bounded if its range is bounded, otherwise, it is unbounded. Thus, a function \(f(x)\) is bounded in the domain \(D\), if there exist two real numbers \(k\) and \(K\) such that $$ k \leq f(x) \leq K \text { for all } x \in D $$ Again, the bounds of the range of a bounded function are called the bounds of the function. Let \(f: X \rightarrow Y\) (i.e., \(f\) is a function whose domain is \(X\) and range \(f(X) \subseteq Y\) the co-domain). \(f\) is called a monotonically increasing function if \(x_{1}, x_{2} \in X\) with \(x_{1}

A function is said to be bounded if its range is bounded, otherwise, it is unbounded. Thus, a function \(f(x)\) is bounded in the domain \(D\), if there exist two real numbers \(k\) and \(K\) such that $$ k \leq f(x) \leq K \text { for all } x \in D $$ Again, the bounds of the range of a bounded function are called the bounds of the function. Let \(f: X \rightarrow Y\) (i.e., \(f\) is a function whose domain is \(X\) and range \(f(X) \subseteq Y\) the co-domain). \(f\) is called a monotonically increasing function if \(x_{1}, x_{2} \in X\) with \(x_{1}

A function is said to be bounded if its range is bounded, otherwise, it is unbounded. Thus, a function \(f(x)\) is bounded in the domain \(D\), if there exist two real numbers \(k\) and \(K\) such that $$ k \leq f(x) \leq K \text { for all } x \in D $$ Again, the bounds of the range of a bounded function are called the bounds of the function. Let \(f: X \rightarrow Y\) (i.e., \(f\) is a function whose domain is \(X\) and range \(f(X) \subseteq Y\) the co-domain). \(f\) is called a monotonically increasing function if \(x_{1}, x_{2} \in X\) with \(x_{1}

A function \(f\) defined on \([a, b]\) is bounded above if there exists a real number \(K\) such that \(|f(x)| \leq K, \forall x \in[a, b]\). The real number \(K\) is called an upper bound of \(f . f\) is called bounded below if there exists a real number \(k\) such that \(|f(x)| \geq k, \forall x \in[a, b]\). The real number \(k\) is called a lower bound of \(f\). It may be observed that if \(K\) is an upper bound of \(f\), then every real number greater than or equal to \(K\) is also an upper bound of \(f\). The smallest of all the upper bounds of \(f\) is called the least upper bound or the supremum (sup.) of \(f\). Similarly, the greatest of all the lower bounds of \(f\) is called the greatest lower bound or the infimum (Inf.) of \(f\). If the functions \(f\) and \(g\) are bounded and continuous on \([a, b]\) and \(g\) keeps the same sign on \([a, b]\), then there exists a number \(\mu\) between the infimum and supremum of \(f\) on \([a, b]\) such that $$ \int_{a}^{b} f(x) g(x) d x=\mu \int_{a}^{b} g(x) d x $$. \(\int_{0}^{1} \frac{\sin \pi x}{1+x^{2}} d x\) 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

A function \(f\) defined on \([a, b]\) is bounded above if there exists a real number \(K\) such that \(|f(x)| \leq K, \forall x \in[a, b]\). The real number \(K\) is called an upper bound of \(f . f\) is called bounded below if there exists a real number \(k\) such that \(|f(x)| \geq k, \forall x \in[a, b]\). The real number \(k\) is called a lower bound of \(f\). It may be observed that if \(K\) is an upper bound of \(f\), then every real number greater than or equal to \(K\) is also an upper bound of \(f\). The smallest of all the upper bounds of \(f\) is called the least upper bound or the supremum (sup.) of \(f\). Similarly, the greatest of all the lower bounds of \(f\) is called the greatest lower bound or the infimum (Inf.) of \(f\). If the functions \(f\) and \(g\) are bounded and continuous on \([a, b]\) and \(g\) keeps the same sign on \([a, b]\), then there exists a number \(\mu\) between the infimum and supremum of \(f\) on \([a, b]\) such that $$ \int_{a}^{b} f(x) g(x) d x=\mu \int_{a}^{b} g(x) d x $$. \(\int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} d x\) 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

A function \(f\) defined on \([a, b]\) is bounded above if there exists a real number \(K\) such that \(|f(x)| \leq K, \forall x \in[a, b]\). The real number \(K\) is called an upper bound of \(f . f\) is called bounded below if there exists a real number \(k\) such that \(|f(x)| \geq k, \forall x \in[a, b]\). The real number \(k\) is called a lower bound of \(f\). It may be observed that if \(K\) is an upper bound of \(f\), then every real number greater than or equal to \(K\) is also an upper bound of \(f\). The smallest of all the upper bounds of \(f\) is called the least upper bound or the supremum (sup.) of \(f\). Similarly, the greatest of all the lower bounds of \(f\) is called the greatest lower bound or the infimum (Inf.) of \(f\). If the functions \(f\) and \(g\) are bounded and continuous on \([a, b]\) and \(g\) keeps the same sign on \([a, b]\), then there exists a number \(\mu\) between the infimum and supremum of \(f\) on \([a, b]\) such that $$ \int_{a}^{b} f(x) g(x) d x=\mu \int_{a}^{b} g(x) d x $$. \(\int_{0}^{1} \frac{x^{2}}{\sqrt{1+x^{2}}}\) 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

We know that for a continuous function \(f\) in \([a, b]\) $$ \lim _{n \rightarrow \infty} \sum_{r=1}^{n} h f(a+r h)=\int_{a}^{b} f(x) d x, h=\frac{b-a}{n} $$ On putting \(a=0, b=1 ;\) (1) becomes $$ \lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n} f\left(\frac{r}{n}\right)=\int_{0}^{1} f(x) d x $$ This formula enables us to evaluate limits of the form: $$ \lim _{n \rightarrow \infty}\left[\phi_{1}(n)+\phi_{2}(n)+\ldots+\phi_{n}(n)\right] $$ To evaluate this limit we express the \(r\) th term as $$ \phi_{r}(n)=\frac{1}{n} f\left(\frac{r}{n}\right) $$ and then replace \(\frac{r}{n}\) by \(x, \frac{1}{n}\) by \(d x\) and \(\lim _{n \rightarrow \infty} \sum_{r=1}^{n}\) by \(\int_{0}^{1}\) Also, \(\lim _{n \rightarrow \infty} \sum_{r=1}^{k n} \frac{1}{n} f\left(\frac{r}{n}\right)=\int_{0}^{k} f(x) d x\) \(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \cdots\left(1+\frac{n}{n}\right)\right]^{1 / n}\) is equal to (A) \(\frac{2}{-}\) (B) \(\frac{e}{2}\) (C) \(\frac{e}{4}\) (D) \(\frac{4}{e}\)

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

Column-I I. \(\int_{0}^{2} x^{3} \sqrt{2 x-x^{2}} d x=\) II. \(\int_{\pi / 2}^{3 \pi / 2}[\sin x] d x=\) (where \([\cdot]\) denotes the greatest integer function) III. \(\int_{0}^{1}|\sin 2 \pi x| d x=\) IV. \(\int_{0}^{\pi} \frac{\sin \left(n+\frac{1}{2}\right) x}{\sin \left(\frac{x}{2}\right)} d x=\) Column-II (A) \(\pi\) (B) \(\frac{7 \pi}{8}\) (C) \(-\frac{\pi}{2}\) (D) \(\frac{2}{\pi}\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion (A) is True and Reason (R) is True; Reason \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) is True; Reason (R) is not a correct explanation for Assertion (A) (C) Assertion (A) is True, Reason (R) is False (D) Assertion (A) is False, Reason (R) is True Assertion: If \(a, b(a

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion (A) is True and Reason (R) is True; Reason \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) is True; Reason (R) is not a correct explanation for Assertion (A) (C) Assertion (A) is True, Reason (R) is False (D) Assertion (A) is False, Reason (R) is True Assertion: \(\int_{0}^{\pi / 6} \frac{\sqrt{3 \cos 2 x-1}}{\cos x} d x\) \(=\sqrt{\frac{2}{3}} \pi-2 \tan ^{-1} \sqrt{2}\) Reason: \(\int_{0}^{\pi / 3} \frac{d x}{5+\cos 2 x}=\frac{1}{2 \sqrt{6}} \tan ^{-1} \sqrt{2}\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion (A) is True and Reason (R) is True; Reason \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) is True; Reason (R) is not a correct explanation for Assertion (A) (C) Assertion (A) is True, Reason (R) is False (D) Assertion (A) is False, Reason (R) is True Assertion: \(\int_{1 / 3}^{11 / 2}\\{x\\} d x=\frac{185}{72}\) (where \(\\{x\\}\) denotes the fractional part of \(x\\}\) Reason: If \(f(x)\) is a periodic function having period, \(T\), then $$ \int_{a}^{b+n T} f(x) d x=n \int_{0}^{T} f(x) d x+\int_{a}^{b} f(x) d x $$

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion (A) is True and Reason (R) is True; Reason \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) is True; Reason (R) is not a correct explanation for Assertion (A) (C) Assertion (A) is True, Reason (R) is False (D) Assertion (A) is False, Reason (R) is True 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 \(1 .\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion (A) is True and Reason (R) is True; Reason \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) is True; Reason (R) is not a correct explanation for Assertion (A) (C) Assertion (A) is True, Reason (R) is False (D) Assertion (A) is False, Reason (R) is True Assertion: \(\int_{0}^{\pi / 2} \sin 2 k x \cot x d x=\frac{\pi}{2}\) Reason: \(\frac{\sin 2 k x}{\sin x}=2(\cos x+\cos 3 x+\ldots+\cos\) \((2 k-1) x)\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion (A) is True and Reason (R) is True; Reason \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) is True; Reason (R) is not a correct explanation for Assertion (A) (C) Assertion (A) is True, Reason (R) is False (D) Assertion (A) is False, Reason (R) is True Assertion: If \(f, g\) and \(h\) be continuous functions on \([0, a]\) such that \(f(x)=f(a-x), g(x)=-g(a-x)\) and \(3 h(x)-4 h(a-x)=5\), then \(\int_{0}^{a} f(x) g(x) h(x) d x=0\) Reason: \(\int_{0}^{a} f(x) g(x) d x=0\)

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