Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.

4851.

\[\int_{\,0}^{\,3}{\,\frac{3x+1}{{{x}^{2}}+9}dx=}\]                                           [EAMCET 2003]

A.                 \[\log (2\sqrt{2})+\frac{\pi }{12}\]           
B.                 \[\log (2\sqrt{2})+\frac{\pi }{2}\]
C.                 \[\log (2\sqrt{2})+\frac{\pi }{6}\]             
D.                 \[\log (2\sqrt{2})+\frac{\pi }{3}\]
Answer» B.                 \[\log (2\sqrt{2})+\frac{\pi }{2}\]
Discussion
4852.

\[\int_{\,0}^{\,1}{\,\sin \left( 2{{\tan }^{-1}}\sqrt{\frac{1+x}{1-x}} \right)\,dx=}\]                                              [EAMCET 2003]

A.                 \[\pi /6\]             
B.                 \[\pi /4\]
C.                 \[\pi /2\]             
D.                 \[\pi \]
Answer» C.                 \[\pi /2\]             
Discussion
4853.

The value of \[\int_{\,0}^{\,\pi }{\,\left| \,{{\sin }^{3}}\theta \, \right|\,d\theta }\] is                      [UPSEAT 2003]

A.                 0             
B.                 3/8
C.                 4/3        
D.                 \[\pi \]
Answer» D.                 \[\pi \]
Discussion
4854.

\[\int_{\,8}^{\,15}{\frac{dx}{(x-3)\sqrt{x+1}}=}\]                                              [UPSEAT 2003]

A.                 \[\frac{1}{2}\log \frac{5}{3}\]     
B.                 \[\frac{1}{3}\log \frac{5}{3}\]
C.                 \[\frac{1}{2}\log \frac{3}{5}\]     
D.                 \[\frac{1}{5}\log \frac{3}{5}\]
Answer» B.                 \[\frac{1}{3}\log \frac{5}{3}\]
Discussion
4855.

\[\int_{1}^{\sqrt{3}}{\frac{1}{1+{{x}^{2}}}dx}\] is equal to                                            [DCE 2002]

A.                 \[\pi /12\]          
B.                 \[\pi /6\]
C.                 \[\pi /4\]             
D.                 \[\pi /3\]
Answer» B.                 \[\pi /6\]
Discussion
4856.

\[\int_{\,2}^{\,3}{\frac{dx}{{{x}^{2}}-x}=}\]                                         [EAMCET 2002]

A.                 \[\log (2/3)\]     
B.                 \[\log (1/4)\]
C.                 \[\log (4/3)\]     
D.                 \[\log (8/3)\]
Answer» D.                 \[\log (8/3)\]
Discussion
4857.

\[\int_{0}^{1}{{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\,dx=}\]                    [Karnataka CET 1999]

A.                 \[\frac{\pi }{2}-2\log \sqrt{2}\]
B.                 \[\frac{\pi }{2}+2\log \sqrt{2}\]
C.                 \[\frac{\pi }{4}-\log \sqrt{2}\]
D.                 \[\frac{\pi }{4}+\log \sqrt{2}\]
Answer» B.                 \[\frac{\pi }{2}+2\log \sqrt{2}\]
Discussion
4858.

The value of \[\int_{\,0}^{\,1}{\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}dx}\] is                               [RPET 2001]

A.                 \[\pi /4\]             
B.                 \[{{\pi }^{2}}/32\]
C.                 1             
D.                 None of these
Answer» C.                 1             
Discussion
4859.

\[\int_{\,-\,1}^{\,0}{\frac{dx}{{{x}^{2}}+2x+2}=}\]                                 [MP PET 2000]

A.                 0             
B.                 \[\pi /4\]
C.                 \[\pi /2\]             
D.                 \[-\pi /4\]
Answer» C.                 \[\pi /2\]             
Discussion
4860.

The value of \[\int_{\,-\,1}^{\,3}{\,{{\tan }^{-1}}\left( \frac{x}{{{x}^{2}}+1} \right)+{{\tan }^{-1}}\left( \frac{{{x}^{2}}+1}{x} \right)\,dx}\] is                 [Karnataka CET 2000]

A.                 \[2\pi \]               
B.                 \[\pi \]
C.                 \[\frac{21}{5}\pi \]          
D.                 \[\frac{\pi }{4}\]
Answer» B.                 \[\pi \]
Discussion
4861.

\[\left( \int_{\,0}^{\,a}{x\,dx} \right)\le (a+4),\] then                                     [RPET 2000]

A.                 \[\frac{24}{5}\pi \]          
B.                 \[-2\le a\le 4\]
C.                 \[-2\le a\le 0\]  
D.                 \[a\le -2\,\,\text{or}\,\,a\ge 4\]
Answer» C.                 \[-2\le a\le 0\]  
Discussion
4862.

\[\int_{0}^{\pi /4}{{}}(\cos x-\sin x)dx+\int_{\pi /4}^{5\pi /4}{{}}(\sin x-\cos x)dx\]                    \[+\int_{2\pi }^{\pi /4}{{}}(\cos x-\sin x)\,dx\] is equal to              [RPET 2000]

A.                 \[\sqrt{2}-2\]    
B.                 \[2\sqrt{2}-2\]
C.                 \[3\sqrt{2}-2\]  
D.                 \[4\sqrt{2}-2\]
Answer» E.
Discussion
4863.

\[\frac{1}{2}(e-3)\]          [SCRA 1986; Karnataka CET 1999]

A.                 \[\pi ab\]            
B.                 \[{{\pi }^{2}}ab\]
C.                 \[\frac{\pi }{ab}\]            
D.                 \[\frac{\pi }{2ab}\]
Answer» E.
Discussion
4864.

\[\int_{\,1}^{\,x}{\frac{\log {{x}^{2}}}{x}\,dx=}\]                                               [DCE 1999]

A.                 \[{{(\log x)}^{2}}\]          
B.                 \[\frac{1}{2}{{(\log x)}^{2}}\]
C.                 \[\frac{\log {{x}^{2}}}{2}\]           
D.                 None of these
Answer» B.                 \[\frac{1}{2}{{(\log x)}^{2}}\]
Discussion
4865.

\[\int_{1}^{e}{\frac{1}{x}\,dx}\] is equals to        [SCRA 1996; Pb. CET 2003]

A.                 \[\infty \]           
B.                 0
C.                 1             
D.                 \[\log (1+e)\]
Answer» D.                 \[\log (1+e)\]
Discussion
4866.

\[\int_{0}^{1}{\sqrt{\frac{1-x}{1+x}}}\,dx\] equals          [RPET 1997; IIT Screening 2004]

A.                 \[\left( \frac{\pi }{2}-1 \right)\] 
B.                 \[\left( \frac{\pi }{2}+1 \right)\]
C.                 \[\frac{\pi }{2}\]              
D.                 \[(\pi +1)\]
Answer» B.                 \[\left( \frac{\pi }{2}+1 \right)\]
Discussion
4867.

\[\int_{0}^{\pi /4}{[\sqrt{\tan x}+\sqrt{\cot x}]\,dx}\] equals                      [RPET 1997]

A.                 \[\sqrt{2}\pi \] 
B.                 \[\frac{\pi }{2}\]
C.                 \[\frac{\pi }{\sqrt{2}}\] 
D.                 \[2\pi \]
Answer» D.                 \[2\pi \]
Discussion
4868.

The value of \[\int_{0}^{1}{\frac{{{x}^{4}}+1}{{{x}^{2}}+1}\,dx}\] is                                          [MP PET 1998]

A.                 \[\frac{1}{6}(3\pi -4)\]  
B.                 \[\frac{1}{6}(3-4\pi )\]
C.                 \[\frac{1}{6}(3\pi +4)\] 
D.                 \[\frac{1}{6}(3+4\pi )\]
Answer» B.                 \[\frac{1}{6}(3-4\pi )\]
Discussion
4869.

\[\int_{1}^{2}{\frac{1}{{{x}^{2}}}{{e}^{\frac{-1}{x}}}\,dx=}\]                                 [DCE 2001]

A.                 \[\sqrt{e}+1\]
B.                 \[\sqrt{e}-1\]
C.                 \[\frac{\sqrt{e}+1}{e}\]
D.                 \[\frac{\sqrt{e}-1}{e}\]
Answer» E.
Discussion
4870.

\[\int_{0}^{\pi /4}{\frac{{{\sec }^{2}}x}{(1+\tan x)(2+\tan x)}}\,dx=\]

A.                 \[{{\log }_{e}}\left( \frac{2}{3} \right)\] 
B.                 \[{{\log }_{e}}3\]
C.                 \[\frac{1}{2}{{\log }_{e}}\left( \frac{4}{3} \right)\]            
D.                 \[{{\log }_{e}}\left( \frac{4}{3} \right)\]
Answer» E.
Discussion
4871.

\[\int_{0}^{2/3}{\frac{dx}{4+9{{x}^{2}}}=}\]                                        [MP PET 1997]

A.                 \[\frac{\pi }{12}\]            
B.                 \[\frac{\pi }{24}\]
C.                 \[\frac{\pi }{4}\]              
D.                 0
Answer» C.                 \[\frac{\pi }{4}\]              
Discussion
4872.

If \[{{I}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\theta \,d\theta ,}\] then \[{{I}_{8}}+{{I}_{6}}\] equals [Kurukshetra CEE 1996]

A.                 \[\frac{1}{4}\]   
B.                 \[\frac{1}{5}\]
C.                 \[\frac{1}{6}\]   
D.                 \[\frac{1}{7}\]
Answer» E.
Discussion
4873.

If for non-zero \[x,\] \[af(x)+bf\left( \frac{1}{x} \right)=\frac{1}{x}-5,\] where \[a\ne b,\] then \[\int_{1}^{2}{f(x)\,dx=}\]                                           [IIT 1996]

A.                 \[\frac{1}{({{a}^{2}}+{{b}^{2}})}\left[ a\log 2-5a+\frac{7}{2}b \right]\]
B.                 \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a+\frac{7}{2}b \right]\]
C.                 \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\]
D.                 \[\frac{1}{({{a}^{2}}+{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\]
Answer» C.                 \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\]
Discussion
4874.

The value of \[\int_{0}^{{{\sin }^{2}}x}{{{\sin }^{-1}}\sqrt{t}\,dt+\int_{0}^{{{\cos }^{2}}x}{{{\cos }^{-1}}\sqrt{t}\,dt}}\] is [MP PET 2001; Orissa JEE 2005]

A.                 \[\frac{\pi }{2}\]              
B.                 1
C.                 \[\frac{\pi }{4}\]              
D.                 None of these
Answer» D.                 None of these
Discussion
4875.

\[\int_{0}^{\pi /4}{\frac{dx}{{{\cos }^{4}}x-{{\cos }^{2}}x{{\sin }^{2}}x+{{\sin }^{4}}x}=}\]

A.                 \[\frac{\pi }{2}\]              
B.                 \[\frac{\pi }{4}\]
C.                 \[\frac{\pi }{3}\]              
D.                 None of these
Answer» B.                 \[\frac{\pi }{4}\]
Discussion
4876.

The value of \[\int_{3}^{5}{\frac{{{x}^{2}}}{{{x}^{2}}-4}\,dx}\] is                                                [Roorkee 1992]

A.                 \[2-{{\log }_{e}}\left( \frac{15}{7} \right)\]
B.                 \[2+{{\log }_{e}}\left( \frac{15}{7} \right)\]
C.                 \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\]
D.                 \[2-{{\tan }^{-1}}\left( \frac{15}{7} \right)\]
Answer» C.                 \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\]
Discussion
4877.

\[\int_{\pi /3}^{\pi /2}{\frac{\sqrt{1+\cos x}}{{{(1-\cos x)}^{\frac{5}{2}}}}}\,dx=\]                                             [AI CBSE 1980]

A.                 \[\frac{5}{2}\]
B.                 \[\frac{3}{2}\]
C.                 \[\frac{1}{2}\]
D.                 \[\frac{2}{5}\]
Answer» C.                 \[\frac{1}{2}\]
Discussion
4878.

The value of \[\int_{1}^{2}{\log x\,dx}\] is                                            [Roorkee 1995]

A.                 \[\log 2/e\]        
B.                 \[\log 4\]
C.                 \[\log 4/e\]        
D.                 \[\log 2\]
Answer» D.                 \[\log 2\]
Discussion
4879.

\[\int_{0}^{\pi /4}{\frac{\sec x}{1+2{{\sin }^{2}}x}}\] is equal to                                 [MNR 1994]

A.                 \[\frac{1}{3}\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\]            
B.                 \[\frac{1}{3}\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\]
C.                 \[3\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\]
D.                 \[3\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\]
Answer» B.                 \[\frac{1}{3}\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\]
Discussion
4880.

The value of \[\int_{0}^{\pi /2}{\frac{\sin x}{1+{{\cos }^{2}}x}\,dx}\] is                   [RPET 1995]

A.                 \[\pi /2\]             
B.                 \[\pi /4\]
C.                 \[\pi /3\]             
D.                 \[\pi /6\]
Answer» C.                 \[\pi /3\]             
Discussion
4881.

The value of \[\int_{0}^{2}{\frac{{{3}^{\sqrt{x}}}}{\sqrt{x}}}\,dx\] is                                        [SCRA 1992]

A.                 \[\frac{2}{\log 3}.({{3}^{\sqrt{2}}}-1)\]  
B.                 0
C.                 \[2.\frac{\sqrt{2}}{\log 3}\]         
D.                 \[\frac{{{3}^{\sqrt{2}}}}{\sqrt{2}}\]
Answer» B.                 0
Discussion
4882.

\[\int_{1/4}^{1/2}{\frac{dx}{\sqrt{x-{{x}^{2}}}}=}\]                                          [SCRA 1986]

A.                 \[\pi \] 
B.                 \[\frac{\pi }{2}\]
C.                 \[\frac{\pi }{3}\]              
D.                 \[\frac{\pi }{6}\]
Answer» E.
Discussion
4883.

If \[x({{x}^{4}}+1)\varphi (x)=1,\] then \[\int_{1}^{2}{\varphi (x)\,dx=}\]              [SCRA 1986]

A.                 \[\frac{1}{4}\log \frac{32}{17}\]
B.                 \[\frac{1}{2}\log \frac{32}{17}\]
C.                 \[\frac{1}{4}\log \frac{16}{17}\]
D.                 None of these
Answer» B.                 \[\frac{1}{2}\log \frac{32}{17}\]
Discussion
4884.

\[\int_{0}^{1}{\frac{{{e}^{x}}(x-1)}{{{(x+1)}^{3}}}\,dx=}\]                                             [SCRA 1986]

A.                 \[\frac{e}{4}\]  
B.                 \[\frac{e}{4}-1\]
C.                 \[\frac{e}{4}+1\]             
D.                 None of these
Answer» C.                 \[\frac{e}{4}+1\]             
Discussion
4885.

\[\int_{0}^{\pi /4}{\frac{4\sin 2\theta \,d\theta }{{{\sin }^{4}}\theta +{{\cos }^{4}}\theta }}=\]                                   [SCRA 1986]

A.                 \[\pi /4\]             
B.                 \[\pi /2\]
C.                 \[\pi \] 
D.                 None of these
Answer» D.                 None of these
Discussion
4886.

\[\int_{0}^{1}{\frac{dx}{\sqrt{1+x}-\sqrt{x}}=}\]                                               [SCRA 1986]

A.                 \[\frac{2\sqrt{2}}{3}\]   
B.                 \[\frac{4\sqrt{2}}{3}\]
C.                 \[\frac{8\sqrt{2}}{3}\]   
D.                 None of these
Answer» C.                 \[\frac{8\sqrt{2}}{3}\]   
Discussion
4887.

\[\int_{0}^{\pi /2}{\frac{\cos x}{(1+\sin x)(2+\sin x)}}\,dx=\]                                 [UPSEAT 1999]

A.                 \[\log \frac{4}{3}\]
B.                 \[\log \frac{1}{3}\]
C.                 \[\log \frac{3}{4}\]
D.                 None of these
Answer» B.                 \[\log \frac{1}{3}\]
Discussion
4888.

If \[\int_{0}^{1}{x\log \left( 1+\frac{x}{2} \right)}\,dx=a+b\log \frac{2}{3},\] then              [SCRA 1986]

A.                 \[a=\frac{3}{2},\,\,\,b=\frac{3}{2}\]        
B.                 \[a=\frac{3}{4},\,\,\,b=-\frac{3}{4}\]
C.                 \[a=\frac{3}{4},\,\,\,b=\frac{3}{2}\]        
D.                 \[a=b\]
Answer» D.                 \[a=b\]
Discussion
4889.

The value of \[\int_{0}^{\pi /4}{\frac{1+\tan x}{1-\tan x}\,dx}\] is                             [SCRA 1986]

A.                 \[-\frac{1}{2}\log 2\]      
B.                 \[\frac{1}{4}\log 2\]
C.                 \[\frac{1}{3}\log 2\]       
D.                 None of these
Answer» B.                 \[\frac{1}{4}\log 2\]
Discussion
4890.

The value of \[\int_{0}^{1}{\frac{dx}{{{e}^{x}}+{{e}^{-x}}}}\] is                                  [SCRA 1980]

A.                 \[{{\tan }^{-1}}\left( \frac{1-e}{1+e} \right)\]     
B.                 \[{{\tan }^{-1}}\left( \frac{e-1}{e+1} \right)\]
C.                 \[\frac{\pi }{4}\]              
D.                 \[{{\tan }^{-1}}e+\frac{\pi }{4}\]
Answer» C.                 \[\frac{\pi }{4}\]              
Discussion
4891.

\[\int_{0}^{\pi /3}{\cos 3x\,dx=}\]                                           [SCRA 1980]

A.                 \[\pi \] 
B.                 0
C.                 \[\frac{\pi }{2}\]              
D.                 \[\frac{\pi }{4}\]
Answer» C.                 \[\frac{\pi }{2}\]              
Discussion
4892.

\[\int_{0}^{1}{{{(1-x)}^{9}}dx=}\]                                             [SCRA 1979]

A.                 1             
B.                 \[\frac{1}{10}\]
C.                 \[\frac{11}{10}\]              
D.                 2
Answer» C.                 \[\frac{11}{10}\]              
Discussion
4893.

\[\int_{0}^{\pi }{\frac{dx}{1-2a\cos x+{{a}^{2}}}}\,\]=                                     [CEE 1993]

A.                 \[\frac{\pi }{2(1-{{a}^{2}})}\]      
B.                 \[\pi (1-{{a}^{2}})\]
C.                 \[\frac{\pi }{1-{{a}^{2}}}\]           
D.                 None of these
Answer» D.                 None of these
Discussion
4894.

\[\int_{-\pi /4}^{\pi /2}{{{e}^{-x}}\sin x\,dx}=\]                                 [CEE 1993]

A.                 \[-\frac{1}{2}{{e}^{-\pi /2}}\]     
B.                 \[-\frac{\sqrt{2}}{2}{{e}^{-\pi /4}}\]
C.                 \[-\sqrt{2}({{e}^{-\pi /4}}+{{e}^{-\pi /4}})\]        
D.                 0
Answer» B.                 \[-\frac{\sqrt{2}}{2}{{e}^{-\pi /4}}\]
Discussion
4895.

\[\int_{0}^{\pi /2}{\frac{1+2\cos x}{{{(2+\cos x)}^{2}}}=}\]                                           [CEE 1993]

A.                 \[\frac{\pi }{2}\]              
B.                 \[\pi \]
C.                 \[\frac{1}{2}\]   
D.                 None of these
Answer» D.                 None of these
Discussion
4896.

If \[{{I}_{1}}=\int_{e}^{{{e}^{2}}}{\frac{dx}{\log x}}\] and \[{{I}_{2}}=\int_{1}^{2}{\frac{{{e}^{x}}}{x}\,dx,}\] then                                                                 [Karnataka CET 2000]

A.                 \[{{I}_{1}}={{I}_{2}}\]    
B.                 \[{{I}_{1}}>{{I}_{2}}\]
C.                 \[{{I}_{1}}<{{I}_{2}}\]    
D.                 None of these
Answer» B.                 \[{{I}_{1}}>{{I}_{2}}\]
Discussion
4897.

\[\int_{1}^{2}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)\,dx=}\] [MNR 1990; AMU 1999; UPSEAT 2000; Pb. CET 2004]

A.                 \[\frac{{{e}^{2}}}{2}+e\]
B.                 \[e-\frac{{{e}^{2}}}{2}\]
C.                 \[\frac{{{e}^{2}}}{2}-e\]
D.                 None of these
Answer» D.                 None of these
Discussion
4898.

The integral \[\int_{-1}^{3}{\left( {{\tan }^{-1}}\frac{x}{{{x}^{2}}+1}+{{\tan }^{-1}}\frac{{{x}^{2}}+1}{x} \right)}\,dx=\]                                                                 [Karnataka CET 2000]

A.                 \[\pi \] 
B.                 \[2\pi \]
C.                 \[3\pi \]               
D.                 None of these
Answer» C.                 \[3\pi \]               
Discussion
4899.

The greater of \[\int_{0}^{\pi /2}{\frac{\sin x}{x}\,dx}\] and \[\frac{\pi }{2},\] is

A.                 \[\frac{\pi }{2}\]              
B.                 \[\int_{0}^{\pi /2}{\frac{\sin x}{x}\,dx}\]
C.                 Nothing can be said       
D.                 None of these
Answer» B.                 \[\int_{0}^{\pi /2}{\frac{\sin x}{x}\,dx}\]
Discussion
4900.

The value of the integral \[\int_{-\pi }^{\pi }{\sin mx\sin nx\,dx}\] for \[m\ne n\] \[(m,\,\,n\in I),\] is

A.                 0             
B.                 \[\pi \]
C.                 \[\frac{\pi }{2}\]              
D.                 \[2\pi \]
Answer» B.                 \[\pi \]
Discussion