Chapter 15

\(\int \frac{\cos 7 x-\cos 8 x}{1+2 \cos 5 x} d x=\) (A) \(\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+c\) (B) \(-\frac{\sin 2 x}{2}-\frac{\sin 3 x}{3}+c\) (C) \(\frac{\sin 2 x}{2}-\frac{\sin 3 x}{3}+c\) (D) none of these.

\(\int \frac{\left(1-\cot ^{n-2} x\right) d x}{\tan x+\cot x \cdot \cot ^{n-2} x}=\) (A) \(\frac{1}{n} \log \left|\sin ^{n} x-\cos ^{n} x\right|+c\) (B) \(\frac{1}{n} \log \left|\sin ^{n} x+\cos ^{n} x\right|+c\) (C) \(\frac{1}{n-1} \log \left|\sin ^{n} x+\cos ^{n} x\right|+c\) (D) none of these

\(\int \frac{(x-1) d x}{(x+1) \sqrt{x^{3}+x^{2}+x}}=k \tan ^{-1} \sqrt{\frac{x^{2}+x+1}{x}}+c\), where \(k=\) (A) 1 (B) 2 (C) 4 (D) none of these

\(\int \frac{d x}{\tan x+\cot x+\sec x+\operatorname{cosec} x}=\) (A) \(\frac{1}{2}(\sin x-\cos x+x)+c\) (B) \(\frac{1}{2}(\sin x-\cos x-x)+c\) (C) \(\frac{1}{2}(\sin x+\cos x+x)+c\) (D) none of these

\(\int \frac{(\sin x-\cos x) d x}{(\sin x+\cos x) \sqrt{\sin x \cos x+\sin ^{2} x \cos ^{2} x}}=\) (A) \(\operatorname{cosec}^{-1}(1+\sin 2 x)+c\) (B) \(-\operatorname{cosec}^{-1}(1+\sin 2 x)+c\) (C) \(\sec ^{-1}(1+\sin 2 x)+c\) (D) \(-\sec ^{-1}(1+\sin 2 x)+c\)

\(\int \frac{d x}{\left(2 a x+x^{2}\right)^{3 / 2}}=\) (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) \(\cos \theta+\sec \theta-1\)

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

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) \(k=-\frac{5}{2}\)

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

Let \(f(x)=\frac{x+2}{2 x+3}\), if \(\int\left(\frac{f(x)}{x^{2}}\right)^{1 / 2} d x\) \(=\frac{1}{\sqrt{2}} g\left(\frac{1+\sqrt{2 f(x)}}{1-\sqrt{2 f(x)}}\right)-\sqrt{\frac{2}{3}} h\left(\frac{\sqrt{3 f(x)}+\sqrt{2}}{\sqrt{3 f(x)-\sqrt{2}}}\right)+C\), then (A) \(g(x)=\log |x|\) (B) \(h(x)=\log |x|\) (C) \(g(x)=\tan ^{-1} x\) (D) \(h(x)=\tan ^{-1} x\)

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) \(f(x)=-2 x\)

\(\int x^{1 / 3}\left(2+x^{2 / 3}\right)^{1 / 4} d x\) 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) \(4 \sqrt{1+\sqrt{x}}+2 \log \left(\frac{\sqrt{1+\sqrt{x}}-1}{\sqrt{1+\sqrt{x}+1}}\right)+C\) (D) none of these

\(\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}} d x\) is equal to (A) \(12\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}+\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (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 \(I_{0}+I_{1}+2\left(I_{2}+\ldots+I_{3}\right)+I_{9}+I_{10}\) 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 In \(=\frac{x^{n-1} \sqrt{x^{2}+a^{2}}}{n}+k \operatorname{In}_{-2}\), 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

(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: \(\int \frac{\sin ^{2} x}{a+b \cos x} d x=\frac{1}{b^{2}}(a x-b \sin x)\) $$ \begin{array}{l}-\frac{2 \sqrt{a^{2}-b^{2}}}{b^{2}} \tan ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \\ \text { Reason: } \int \frac{d x}{a+b \cos x} \\ \quad=\frac{2}{\sqrt{a^{2}-b^{2}}} \tan ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right)\end{array} $$

\(\int \frac{d x}{x\left(x^{n}+1\right)}\) is equal to : (A) \(\frac{1}{n} \log \left(\frac{x^{n}}{x^{x}+1}\right)+c\) (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 (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) \(\frac{1}{2} \operatorname{logtan}\left(\frac{x}{2}+\frac{\pi}{12}\right)+\mathrm{c}\) (B) \(\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}\) (C) \(\log \tan \left(\frac{-}{2}-\frac{\underline{\phantom{xx}}}{12}\right)+\mathrm{c}\) (D) \(\log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}\)

The value of \(\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right) \text { is }}\) (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 \(\mathrm{y}(0)=2\), 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) \(\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-3 \int x^{3} \Psi\left(x^{3}\right) d x+C\) [2013] (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: (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: (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) \(\frac{-x^{5}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C\)