MCQOPTIONS
Saved Bookmarks
MCQOPTIONS
This section includes 26 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
\[\int{\frac{dx}{{{a}^{2}}-{{x}^{2}}}}\] is equal to[EAMCET 2002] |
| A. | \[\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)\] |
| B. | \[\frac{1}{2a}{{\sin }^{-1}}\left( \frac{a-x}{a+x} \right)\] |
| C. | \[\frac{1}{2a}\log \,\left( \frac{a+x}{a-x} \right)\] |
| D. | \[\frac{1}{2a}\log \,\left( \frac{a-x}{a+x} \right)\] |
| Answer» D. \[\frac{1}{2a}\log \,\left( \frac{a-x}{a+x} \right)\] | |
| 2. |
\[\int_{{}}^{{}}{\frac{dx}{4{{x}^{2}}+9}=}\] [MP PET 1991; Roorkee 1977; MNR 1974] |
| A. | \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{2x}{3} \right)+c\] |
| B. | \[\frac{3}{2}{{\tan }^{-1}}\left( \frac{2x}{3} \right)+c\] |
| C. | \[\frac{1}{6}{{\tan }^{-1}}\left( \frac{2x}{3} \right)+c\] |
| D. | \[\frac{1}{6}{{\tan }^{-1}}\left( \frac{3x}{2} \right)+c\] |
| Answer» D. \[\frac{1}{6}{{\tan }^{-1}}\left( \frac{3x}{2} \right)+c\] | |
| 3. |
\[\int{\frac{{{(x+1)}^{2}}\,\,dx}{x({{x}^{2}}+1)}}\] is equal to [MP PET 2003] |
| A. | \[{{\log }_{e}}x+c\] |
| B. | \[{{\log }_{e}}x+2{{\tan }^{-1}}x+c\] |
| C. | \[{{\log }_{e}}\frac{1}{{{x}^{2}}+1}+c\] |
| D. | \[{{\log }_{e}}\{x({{x}^{2}}+1)\}+c\] |
| Answer» C. \[{{\log }_{e}}\frac{1}{{{x}^{2}}+1}+c\] | |
| 4. |
\[\int_{{}}^{{}}{\frac{\cos 2x-\cos 2\alpha }{\cos x-\cos \alpha }}dx=\] [MP PET 1994] |
| A. | \[2[\sin x+x\cos \alpha ]+c\] |
| B. | \[2[\sin x+\sin \alpha ]+c\] |
| C. | \[2[-\sin x+x\cos \alpha ]+c\] |
| D. | \[-2[\sin x+\sin \alpha ]+c\] |
| Answer» B. \[2[\sin x+\sin \alpha ]+c\] | |
| 5. |
\[\int_{{}}^{{}}{\frac{dx}{{{\sin }^{2}}x{{\cos }^{2}}x}=}\] [Roorkee 1976; RPET 1991] |
| A. | \[\tan x+\cot x+c\] |
| B. | \[\cot x-\tan x+c\] |
| C. | \[\tan x-\cot x+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 6. |
\[\int{\frac{dx}{\sin x-\cos x+\sqrt{2}}}\] equals[MP PET 2002] |
| A. | \[-\frac{1}{\sqrt{2}}\tan \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] |
| B. | \[\frac{1}{\sqrt{2}}\tan \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] |
| C. | \[\frac{1}{\sqrt{2}}\cot \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] |
| D. | \[-\frac{1}{\sqrt{2}}\cot \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] |
| Answer» E. | |
| 7. |
\[\int_{{}}^{{}}{\frac{{{x}^{5}}}{\sqrt{1+{{x}^{3}}}}dx=}\][IIT 1985] |
| A. | \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}+c\] |
| B. | \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}+\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\] |
| C. | \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}-\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 8. |
\[\int_{{}}^{{}}{\frac{dx}{(1+{{x}^{2}})\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}}}=\] |
| A. | \[\frac{1}{q}\log [q{{\tan }^{-1}}x+\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}]+c\] |
| B. | \[\log [q{{\tan }^{-1}}x+\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}]+c\] |
| C. | \[\frac{2}{3q}{{({{p}^{2}}+{{q}^{2}}{{\tan }^{-1}}x)}^{3/2}}+c\] |
| D. | None of these |
| Answer» B. \[\log [q{{\tan }^{-1}}x+\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}]+c\] | |
| 9. |
\[\int_{{}}^{{}}{\frac{{{x}^{3}}-x-2}{(1-{{x}^{2}})}\ dx=}\] [AI CBSE 1985] |
| A. | \[\log \left( \frac{x+1}{x-1} \right)-\frac{{{x}^{2}}}{2}+c\] |
| B. | \[\log \left( \frac{x-1}{x+1} \right)+\frac{{{x}^{2}}}{2}+c\] |
| C. | \[\log \left( \frac{x+1}{x-1} \right)+\frac{{{x}^{2}}}{2}+c\] |
| D. | \[\log \left( \frac{x-1}{x+1} \right)-\frac{{{x}^{2}}}{2}+c\] |
| Answer» E. | |
| 10. |
If \[\int_{{}}^{{}}{(\sin 2x+\cos 2x)\ dx=\frac{1}{\sqrt{2}}\sin (2x-c)+a}\], then the value of a and c is [Roorkee 1978] |
| A. | \[c=\pi /4\] and \[a=k\] (an arbitrary constant) |
| B. | \[c=-\pi /4\] and \[a=\pi /2\] |
| C. | \[c=\pi /2\] and a is an arbitrary constant |
| D. | None of these |
| Answer» B. \[c=-\pi /4\] and \[a=\pi /2\] | |
| 11. |
\[\int_{{}}^{{}}{\frac{3\cos x+3\sin x}{4\sin x+5\cos x}\ dx=}\][EAMCET 1991] |
| A. | \[\frac{27}{41}x-\frac{3}{41}\log (4\sin x+5\cos x)\] |
| B. | \[\frac{27}{41}x+\frac{3}{41}\log (4\sin x+5\cos x)\] |
| C. | \[\frac{27}{41}x-\frac{3}{41}\log (4\sin x-5\cos x)\] |
| D. | None of these |
| Answer» B. \[\frac{27}{41}x+\frac{3}{41}\log (4\sin x+5\cos x)\] | |
| 12. |
If \[\int_{{}}^{{}}{\frac{2x+3}{(x-1)({{x}^{2}}+1)}\ dx={{\log }_{e}}\left\{ {{(x-1)}^{\frac{5}{2}}}{{({{x}^{2}}+1)}^{a}} \right\}}-\frac{1}{2}{{\tan }^{-1}}x+A\], where A is any arbitrary constant, then the value of ?a? is[MP PET 1998] |
| A. | 44291 |
| B. | -1.66666666666667 |
| C. | -0.833333333333333 |
| D. | -1.25 |
| Answer» E. | |
| 13. |
\[\int_{{}}^{{}}{\frac{dx}{(\sin x+\sin 2x)}=}\] [IIT 1984] |
| A. | \[\frac{1}{6}\log (1-\cos x)+\frac{1}{2}\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] |
| B. | \[6\log (1-\cos x)+2\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] |
| C. | \[6\log (1-\cos x)+\frac{1}{2}\log (1+\cos x)+\frac{2}{3}\log (1+2\cos x)\] |
| D. | None of these |
| Answer» B. \[6\log (1-\cos x)+2\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] | |
| 14. |
\[\int_{{}}^{{}}{\frac{x}{{{x}^{4}}+{{x}^{2}}+1}dx}\] equal to [MP PET 2004] |
| A. | \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{2{{x}^{2}}+1}{3} \right)\] |
| B. | \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{2{{x}^{2}}+1}{\sqrt{3}} \right)\] |
| C. | \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}(2{{x}^{2}}+1)\] |
| D. | None of these |
| Answer» C. \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}(2{{x}^{2}}+1)\] | |
| 15. |
\[\int_{{}}^{{}}{\frac{x-1}{{{(x+1)}^{3}}}{{e}^{x}}\ dx=}\] [IIT 1983; MP PET 1990] |
| A. | \[\frac{-{{e}^{x}}}{{{(x+1)}^{2}}}+c\] |
| B. | \[\frac{{{e}^{x}}}{{{(x+1)}^{2}}}+c\] |
| C. | \[\frac{{{e}^{x}}}{{{(x+1)}^{3}}}+c\] |
| D. | \[\frac{-{{e}^{x}}}{{{(x+1)}^{3}}}+c\] |
| Answer» C. \[\frac{{{e}^{x}}}{{{(x+1)}^{3}}}+c\] | |
| 16. |
If \[\int_{{}}^{{}}{\frac{4{{e}^{x}}+6{{e}^{-x}}}{9{{e}^{x}}-4{{e}^{-x}}}dx=Ax+B\log (9{{e}^{2x}}-4)}+C\], then A, B and C are [IIT 1990] |
| A. | \[A=\frac{3}{2},\ B=\frac{36}{35},\ C=\frac{3}{2}\log 3+\]constant |
| B. | \[A=\frac{3}{2},\ B=\frac{35}{36},\ C=\frac{3}{2}\log 3+\]constant |
| C. | \[A=-\frac{3}{2},\ B=-\frac{35}{36},\ C=-\frac{3}{2}\log 3+\]constant |
| D. | None of these |
| Answer» E. | |
| 17. |
\[\int_{{}}^{{}}{\frac{{{x}^{2}}+1}{{{x}^{4}}-{{x}^{2}}+1}\ dx=}\] [MP PET 1991] |
| A. | \[{{\tan }^{-1}}\left( \frac{1+{{x}^{2}}}{x} \right)+c\] |
| B. | \[{{\cot }^{-1}}\left( \frac{1+{{x}^{2}}}{x} \right)+c\] |
| C. | \[{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] |
| D. | \[{{\cot }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] |
| Answer» D. \[{{\cot }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] | |
| 18. |
\[\int_{{}}^{{}}{\frac{dx}{4{{\sin }^{2}}x+5{{\cos }^{2}}x}=}\] [AISSE 1986] |
| A. | \[\frac{1}{\sqrt{5}}{{\tan }^{-1}}\left( \frac{2\tan x}{\sqrt{5}} \right)+c\] |
| B. | \[\frac{1}{\sqrt{5}}{{\tan }^{-1}}\left( \frac{\tan x}{\sqrt{5}} \right)+c\] |
| C. | \[\frac{1}{2\sqrt{5}}{{\tan }^{-1}}\left( \frac{2\tan x}{\sqrt{5}} \right)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 19. |
\[\int_{{}}^{{}}{\frac{{{x}^{2}}}{{{(9-{{x}^{2}})}^{3/2}}}\ dx=}\] |
| A. | \[\frac{x}{\sqrt{9-{{x}^{2}}}}-{{\sin }^{-1}}\frac{x}{3}+c\] |
| B. | \[\frac{x}{\sqrt{9-{{x}^{2}}}}+{{\sin }^{-1}}\frac{x}{3}+c\] |
| C. | \[{{\sin }^{-1}}\frac{x}{3}-\frac{x}{\sqrt{9-{{x}^{2}}}}+c\] |
| D. | None of these |
| Answer» B. \[\frac{x}{\sqrt{9-{{x}^{2}}}}+{{\sin }^{-1}}\frac{x}{3}+c\] | |
| 20. |
\[\int_{{}}^{{}}{\frac{a\ dx}{b+c{{e}^{x}}}}=\][MP PET 1988; BIT Ranchi 1979] |
| A. | \[\frac{a}{b}\log \left( \frac{{{e}^{x}}}{b+c{{e}^{x}}} \right)+c\] |
| B. | \[\frac{a}{b}\log \left( \frac{b+c{{e}^{x}}}{{{e}^{x}}} \right)+c\] |
| C. | \[\frac{b}{a}\log \left( \frac{{{e}^{x}}}{b+c{{e}^{x}}} \right)+c\] |
| D. | \[\frac{b}{a}\log \left( \frac{b+c{{e}^{x}}}{{{e}^{x}}} \right)+c\] |
| Answer» B. \[\frac{a}{b}\log \left( \frac{b+c{{e}^{x}}}{{{e}^{x}}} \right)+c\] | |
| 21. |
\[\int_{{}}^{{}}{{{\tan }^{2}}x\ dx}\] is equal to [SCRA 1996] |
| A. | \[\tan x+x+c\] |
| B. | \[\tan x-x+c\] |
| C. | \[\sec x+x+c\] |
| D. | \[\sec x-x+c\] |
| Answer» C. \[\sec x+x+c\] | |
| 22. |
\[\int_{{}}^{{}}{\sqrt{1+\sin x}\ dx=}\] [MP PET 1995] |
| A. | \[\frac{1}{2}\left( \sin \frac{x}{2}+\cos \frac{x}{2} \right)+c\] |
| B. | \[\frac{1}{2}\left( \sin \frac{x}{2}-\cos \frac{x}{2} \right)+c\] |
| C. | \[2\sqrt{1+\sin x}+c\] |
| D. | \[-2\sqrt{1-\sin x}+c\] |
| Answer» E. | |
| 23. |
If \[\int_{{}}^{{}}{(\cos x-\sin x)\ dx=\sqrt{2}\sin (x+\alpha )+c}\], then \[\alpha =\] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[-\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | \[-\frac{\pi }{4}\] |
| Answer» D. \[-\frac{\pi }{4}\] | |
| 24. |
The value of \[\int{{{\sec }^{3}}x\,\,dx}\] will be[UPSEAT 1999] |
| A. | \[\frac{1}{2}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] |
| B. | \[\frac{1}{3}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] |
| C. | \[\frac{1}{4}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] |
| D. | \[\frac{1}{8}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] |
| Answer» B. \[\frac{1}{3}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] | |
| 25. |
\[\int_{{}}^{{}}{{{(\log x)}^{2}}\ dx=}\][IIT 1971, 77] |
| A. | \[x{{(\log x)}^{2}}-2x\log x-2x+c\] |
| B. | \[x{{(\log x)}^{2}}-2x\log x-x+c\] |
| C. | \[x{{(\log x)}^{2}}-2x\log x+2x+c\] |
| D. | \[x{{(\log x)}^{2}}-2x\log x+x+c\] |
| Answer» D. \[x{{(\log x)}^{2}}-2x\log x+x+c\] | |
| 26. |
\[\int_{{}}^{{}}{\sin \sqrt{x}}\ dx=\] [Roorkee 1977] |
| A. | \[2[\sin \sqrt{x}-\cos \sqrt{x}]+c\] |
| B. | \[2[\sin \sqrt{x}-\sqrt{x}\cos \sqrt{x}]+c\] |
| C. | \[2[\sin \sqrt{x}+\cos \sqrt{x}]+c\] |
| D. | \[2[\sin \sqrt{x}+\sqrt{x}\cos \sqrt{x}]+c\] |
| Answer» C. \[2[\sin \sqrt{x}+\cos \sqrt{x}]+c\] | |