Chapter 15

\(\int \frac{d x}{\left(2 a x+x^{2}\right)^{1 / 2}}=\) (A) \(\frac{1}{a^{2}} \frac{x+a}{\sqrt{x^{2}+2 a x}}+c\) (B) \(\frac{1}{a^{2}} \frac{x-a}{\sqrt{x^{2}+2 a x}}+c\) (C) \(-\frac{1}{a^{2}} \cdot \frac{x+a}{\sqrt{x^{2}+2 a x}}+c\) (D) none of these

\(\int \frac{\sin ^{3} \theta / 2}{\cos \theta / 2 \sqrt{\cos ^{3} \theta+\cos ^{2} \theta+\cos \theta}} d \theta\) \(=\tan ^{-1} \sqrt{k}+C\), where \(k=\) (A) \(\cos \theta+\sec \theta+1\) (B) \(\cos \theta-\sec \theta+1\) (C) \(\cos \theta+\sec \theta-1\) (D) none of these

\(\int \frac{(2 \sin \theta+\sin 2 \theta) d \theta}{(\cos \theta-1) \sqrt{\cos \theta+\cos ^{2} \theta+\cos ^{3} \theta}}\) \(=-\frac{2}{3} \log \left|\frac{k-\sqrt{3}}{k+\sqrt{3}}\right|+c\), where \(k=\) (A) \(\sqrt{\cos \theta+\sec \theta+1}\) (B) \(\cos \theta+\sec \theta+1\) (C) \(\sqrt{\cos \theta+\sec \theta-1}\) (D) \(\cos \theta+\sec \theta-1\)

If \(f(x)=\lim _{n \rightarrow \infty}\left[2 x+4 x^{3}+\ldots+2 n x^{2 n-1}\right], 0

\(\int \frac{d x}{(1+\sqrt{x})^{8}}=\frac{-1}{3(1+\sqrt{x})^{k_{1}}}+\frac{2}{7(1+\sqrt{x})^{k_{3}}}+C\) where (A) \(k_{1}=6\) (B) \(k_{2}=7\) (C) \(k_{1}=-6\) (D) \(k_{2}=-7\)

If \(\int f(x) \sin 2 x d x=\frac{\ln |f(x)|}{b^{2}-a^{2}}+C\), then \(f(x)\) is equal (A) \(\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}\) (B) \(\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}\) (C) \(\frac{1}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}\) (D) \(\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}\)

If \(\int \frac{(\sqrt{x})^{5}}{(\sqrt{x})^{7}+x^{6}} d x=a \log \left(\frac{x^{k}}{1+x^{k}}\right)+C\), then (A) \(a=\frac{2}{5}\) (B) \(a=-\frac{2}{5}\) (C) \(k=\frac{5}{2}\) (D) \(k=-\frac{5}{2}\)

If \(P=\int e^{a x} \cos b x d x\) and \(Q=\int e^{\alpha x} \sin b x d x\), then (A) \(P=\frac{e^{a x}}{\sqrt{a^{2}+b^{2}}} \cos \left(b x-\tan ^{-1} \frac{b}{a}\right)\) (B) \(Q=\frac{e^{a x}}{\sqrt{a^{2}+b^{2}}} \sin \left(b x-\tan ^{-1} \frac{b}{a}\right)\) (C) \(\left(P^{2}+Q^{2}\right)\left(a^{2}+b^{2}\right)=e^{2} a x\) (D) \(\tan ^{-1}\left(\frac{Q}{P}\right)+\tan ^{-1} \frac{b}{a}=b x\)

\(\int \frac{x^{4}+1}{x^{6}+1} d x=\tan ^{-1} k_{1}-\frac{2}{3} \tan ^{-1} k_{2}+C\), where (A) \(k_{1}=x+\frac{1}{x}\) (B) \(k_{2}=x^{3}\) (C) \(k_{1}=x-\frac{1}{v}\) (D) \(k_{2}=x^{4}\)

If \(\int \frac{x \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x\) \(=A \sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)+B x+C\), then (A) \(A=-1\) (B) \(B=-1\) (C) \(A=1\) (D) none of these

If \(\int \frac{3 \cot 3 x-\cot x}{\tan x-3 \tan 3 x} d x=A x+B \log \left|\frac{\sqrt{3}-\tan x}{\sqrt{3}+\tan x}\right|+\) \(C\), then (A) \(A=1\) (B) \(B=-\sqrt{3}\) (C) \(B=-\frac{1}{\sqrt{3}}\) (D) none of these

If for all \(x \in[-1,0), \int\left(\cos ^{-1} x+\cos ^{-1} \sqrt{1-x^{2}}\right) d x\) \(=A x+f(x) \sin ^{-1} x-2 \sqrt{1-x^{2}}+C\), then (A) \(A=\frac{\pi}{4}\) (B) \(A=\frac{\pi}{2}\) (C) \(f(x)=x\) (D) \(f(x)=-2 x\)

If \(\int \frac{x^{2}+20}{(x \sin x+5 \cos x)^{2}} d x=-\frac{x}{A \cos x}+B\), then (A) \(A=x \sin x+5 \cos x\) (B) \(B=\cot x\) (C) \(A=-(x \sin x+5 \cos x)\) (D) \(B=\tan x\)

\(\int x^{1 / 3}\left(2+x^{2 / 3}\right)^{1 / 4} d x\) is equal to (A) \(\frac{2}{3}\left(2+x^{2 / 3}\right)^{9 / 4}+\frac{12}{5}\left(2+x^{2 / 3}\right)^{5 / 4}+C\) (B) \(\frac{2}{3}\left(2+x^{2 / 3}\right)^{9 / 4}-\frac{12}{5}\left(2+x^{2 / 3}\right)^{5 / 4}+C\) (C) \(\frac{1}{3}\left(2+x^{2 / 3}\right)^{9 / 4}-\frac{12}{5}\left(2+x^{2 / 3}\right)^{5 / 4}+C\) (D) none of these

\(\int \frac{\sqrt{1+\sqrt{x}}}{x} d x\) is equal to (A) \(2 \sqrt{1+\sqrt{x}}-2 \log \left(\frac{\sqrt{1+\sqrt{x}}-1}{\sqrt{1+\sqrt{x}}+1}\right)+C\) (B) \(4 \sqrt{1+\sqrt{x}}+2 \log \left(\frac{\sqrt{1+\sqrt{x}}+1}{\sqrt{1+\sqrt{x}}-1}\right)+C\) (C) \(\sqrt[4]{1+\sqrt{x}}+2 \log \left(\frac{\sqrt{1+\sqrt{x}}-1}{\sqrt{1+\sqrt{x}}+1}\right)+C\) (D) none of these

\(\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}} d x\) is equal to (A) \(12\left(\frac{(1+\sqrt[4]{x})^{n / 3}}{7}+\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (B) \(12\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}-\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (C) \(6\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}-\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (D) none of these

\(\int \sqrt[3]{x} \sqrt[7]{1+\sqrt[3]{x^{4}}} d x\) is equal to (A) \(\frac{21}{32}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C\) (B) \(\frac{32}{21}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C\) (C) \(\frac{7}{32}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C\) (D) none of these

If \(I n=\int \tan n x d x\), then \(I_{0}+I_{1}+2\left(I_{2}+\ldots+I_{n}\right)+I_{0}+I_{10}\) is equal to (A) \(\left(\frac{\tan x}{1}+\frac{\tan ^{2} x}{2}+\ldots+\frac{\tan ^{9} x}{9}\right)\) (B) \(-\left(\frac{\tan x}{1}+\frac{\tan ^{2} x}{2}+\ldots+\frac{\tan ^{9} x}{9}\right)\) (C) \(\left(\frac{\cot x}{1}+\frac{\cot ^{2} x}{2}+\ldots+\frac{\cot ^{9} x}{9}\right)\) (D) \(-\left(\frac{\cot x}{1}+\frac{\cot ^{2} x}{2}+\ldots+\frac{\cot ^{9} x}{9}\right)\)

If \(I n=\int \frac{x^{n}}{\sqrt{x^{2}+a^{2}}} d x(n \geq 2)\), then \(I n=\frac{x^{n-1} \sqrt{x^{2}+a^{2}}}{n}+k I n_{-2}\), where \(k=\) (A) \(\frac{a^{2}(1-n)}{n} I_{n-2}\) (B) \(\frac{a^{2}(n-1)}{n} I_{n-2}\) (C) \(\frac{a^{2}(n+1)}{n} I_{n-2}\) (D) none of these

Column-I Column-II (A) \(\int \sqrt{\sec x-1} d x\) 1\. \(\sin ^{-1}(\tan x)\) (B) \(\int \frac{d x}{\cos x \sqrt{\cos 2 x}}\) 2\. \(\sec ^{-1}(\sec x+\cos x)\) (C) \(\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}\) 3\. \(-2 \log \left(\cos \frac{x}{2}+\sqrt{\cos ^{2} \frac{x}{2}-\frac{1}{2}}\right)\) (D) \(\int \frac{\sin ^{3} x d x}{\left(1+\cos ^{2} x\right) \sqrt{1+\cos ^{2} x+\cos ^{4} x}}\) 4\. \(\sqrt{2}\left(\sqrt{\tan x}+\frac{1}{5} \tan ^{5 / 2} x\right)\)

Column-I Column-II (A) \(\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^{2}}}\) 1\. \(\frac{2(\sqrt{x}-1)}{\sqrt{1-x}}\) (B) \(\int\left(\frac{1-\sqrt{x}}{1+\sqrt{x}}\right)^{1 / 2} \frac{d x}{x}\) 2\. \(\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^{2}+1}{\sqrt{2} x}\right)\) (C) \(\int x \sqrt{\frac{1+x}{1-x}} d x\) 3\. \(2 \cot ^{-1} \sqrt{x}-2 \ln \left|\frac{1+\sqrt{1-x}}{\sqrt{x}}\right|\) (D) \(\int \frac{\left(x^{2}-1\right)}{\left(x^{2}+1\right) \sqrt{x^{4}+1}}\) 4\. \(-\left(1+\frac{x}{2}\right) \sqrt{1-x^{2}}-\frac{1}{2} \cos ^{-1} x\)

Assertion: \(\int e^{(x \sin x+\cos x)}\left(\frac{x^{2} \cos ^{2} x-x \sin x-\cos x}{x^{2}}\right) d x\) \(=e^{(x \sin x+\cos x)} \cdot \frac{\cos x}{x}+C\) Reason: \(\int e^{g(x)}\left\\{f(x) g^{\prime}(x)+f^{\prime}(x)\right\\} d x\) \(=e g^{\prime} x^{\prime} f(x)\)

Assertion: \(\begin{aligned} \int x^{x+1} \log x(1+\log x) d x \\ &=x^{x}(x \log x-1)+C \end{aligned}\) Reason: \(\int x^{x}(1+\log x) d x=x^{x}\)

\(\int \frac{d x}{x\left(x^{n}+1\right)}\) is equal to: (A) \(\frac{1}{n} \log \left(\frac{x^{\prime \prime}}{x^{n}+1}\right)+c\) (B) \(\frac{1}{n} \log \left(\frac{x^{n}+1}{x^{n}}\right)+c\) (C) \(\log \left(\frac{x^{n}}{x^{n}+1}\right)+c\) (D) none of these

The coefficient of the middle term in the binomial expansion in powers of \(x\) of \((1+\alpha x)^{4}\) and of \((1-\alpha x)\) is the same if \(\alpha\) equals \([2004]\) (A) \(-\frac{5}{3}\) (B) \(\frac{3}{5}\) (C) \(-\frac{3}{10}\) (D) \(\frac{10}{3}\)

\(\int\left\\{\frac{(\log x-1)}{\left(1+(\log x)^{2}\right.}\right\\} d x\) is equal to (A) \(\frac{\log x}{(\log x)^{2}+1}+C\) (B) \(\frac{x}{x^{2}+1}+C\) (C) \(\frac{x e^{x}}{1+x^{2}}+C\) (D) \(\frac{x}{(\log x)^{2}+1}+C\)

\(\int \frac{d x}{\cos x+\sqrt{3} \sin x}\) equals (A) \(\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+\mathrm{c}\) (B) \(\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}\) (C) \(\operatorname{logtan}\left(\frac{-}{2}-\frac{\underline{\phantom{xx}}}{12}\right)+\mathrm{c}\) (D) \(\log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}\)

The value of \(\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}\) is (A) \(x+\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c \mathrm{x}\) (B) \(x-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\) (C) \(x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\) (D) \(x-\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c\)

If \(\frac{d y}{d x}=y+3 ; y>-3\) and \(y(0)=2\), then y \((\ln 2)\) is equal to (A) 5 (B) 13 (C) 2 (D) 7

If the integral \(\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k\) then a is equal to (A) \(-1\) (B) \(-2\) (C) 1 (D) 2

If \(\int f(x) d x=\Psi(x)\) then \(\int x^{5} f\left(x^{3}\right) d x\) is equal to (A) \(\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-3 \int x^{3} \Psi\left(x^{3}\right) d x+C\) \([\mathbf{2 0 1 3}]\) (B) \(\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x+C\) (C) \(\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{3} \Psi\left(x^{3}\right) d x\right]+C\) (D) \(\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x\right]+C\)

The integral \(\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x\) is equal to \([2014]\) (A) \((x-1) e^{x+\frac{1}{x}}+c\) (B) \(x e^{x+\frac{1}{x}}+c\) (C) \((x+1) e^{x+\frac{1}{x}}+c\) (D) \(-x e^{x+\frac{1}{x}}+c\)

The integral \(\int \frac{d x}{x^{2}\left(x^{4}+1\right)^{3 / 4}}\) equals: [2015] (A) \(\left(x^{4}+1\right)^{1 / 4}+c\) (B) \(-\left(x^{4}+1\right)^{1 / 4}+c\) (C) \(-\left(\frac{x^{4}+1}{x^{4}}\right)^{1 / 4}+c\) (D) \(\left(\frac{x^{4}+1}{x^{4}}\right)^{1 / 4}+c\)

The integral \(\int \frac{2 x^{12}+5 x^{9}}{\left(x^{5}+x^{3}+1\right)}\) is equals to: \(\quad\) [2016] (A) \(\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)}+C\) (B) \(\frac{-x^{5}}{\left(x^{5}+x^{3}+1\right)^{2}}+C\) (C) \(\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C\) (D) \(\frac{-x^{5}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C\)