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Q39  Choose the correct answer  \int \frac{dx }{\sin ^ 2 x \cos ^2 x }\: \: \: equals 

 (A) \tan x + \cot x + C \\\\ (B) \tan x - \cot x + C\\\\ (C) \tan x \cot x + C\\\\ (D) \tan x - \cot 2x + C

Given integral                                            Therefore, the correct answer is B.  

Q38  Choose the correct answer \int \frac{10 x^ 9 + 10 ^x \log _ e 10 dx }{x ^{10}+ 10 ^x }dx\: \: \: equals

(A) 10^x - x^{10} + C \\\\(B) 10^x + x^{10} + C\\\\ (C) (10^x - x^{10})^{-1} + C \\\\ (D) log (10^x + x^{10}) + C

 

Given integral  Taking the denominator  Now differentiating both sides we get Back substituting the value of t, Therefore the correct answer is D.

Q37  Integrate the functions  \frac{x ^3 \sin ( \tan ^{-1} x ^ 4 )}{1 + x ^8 }

Given function   Assume that              ......................(1) Now to solve further we take  So, from the equation (1), we will get Now back substitute the value of u, and then back substituting the value of t,

Q36  Integrate the functions  \frac{( x+1)( x+ \log x )^2}{x }

Given function   Simplifying to solve easier;                                           Assume that  Now, back substituting the value of t,  

Q35  Integrate the functions  \frac{( 1+ \log x )^2}{x}

Given function   Assume that  Now, back substituting the value of t,    

Q34  Integrate the functions  \frac{\sqrt { \tan x } }{\sin x \cos x }

Given function   Assume that   Now solving the assumed integral; Multiplying numerator and denominator by ; Now, to solve further we will assume  Or,  Now, back substituting the value of t,  

Q33  Integrate the functions  \frac{1}{1- \tan x }

Given function   Assume that   Now solving the assumed integral; Now, to solve further we will assume  Or,  Now, back substituting the value of t,  

32) Integrate the functions  \frac{1}{1+ \cot x }

Given function   Assume that   Now solving the assumed integral; Now, to solve further we will assume  Or,  Now, back substituting the value of t,    

Q31  Integrate the functions \frac{\sin x }{( 1+ \cos x )^2}

Given function  , Assume the    Now, back substituted the value of t. , where C is any constant value.

Q 30  Integrate the functions  \frac{\sin x }{1+ \cos x }

Given function  , Assume the    Now, back substituted the value of t. , where C is any constant value.

Q29  Integrate the functions \cot x \: log \sin x

Given function  , Assume the    Now, back substituted the value of t. , where C is any constant value.

Q28  Integrate the functions \frac{\cos x }{\sqrt { 1+ \sin x }}

Given function  , Assume the    Now, back substituted the value of t.  , where C is any constant value.

Q27  Integrate the functions \sqrt { \sin 2x } \cos 2x

Given function  , Assume the    Now, back substituted the value of t.  , where C is any constant value.

Q26  Integrate the functions \frac{\cos \sqrt x }{\sqrt x }

Given function  , Assume the    Now, back substituted the value of t. , where C is any constant value.

 Q25  Integrate the functions \frac{1 }{ \cos ^2 x (1-\tan x )^2}

Given function  , or simplified as  Assume the    Now, back substituted the value of t.  where C is any constant value.

Q24  Integrate the functions  \frac{2 \cos x - 3\sin x }{6 \cos x + 4 \sin x }

Given function  , or simplified as  Assume the    Now, back substituted the value of t. , where C is any constant value.

 Q23  Integrate the functions \frac{\sin ^{-1}x}{\sqrt { 1- x^2 }}

Given function  , Assume the                           Now, back substituted the value of t. , where C is any constant value.

Q22  Integrate the functions  \sec ^2 ( 7- 4x )

Given function  , Assume the                           Now, back substituted the value of t. , where C is any constant value.

Q21  Integrate the functions  \tan ^2 ( 2x-3 )

Given function  , Assume the                          Now, back substituting the value of t, or  , where C is any constant value.

 Q20   Integrate the functions \frac{e ^{2x}- e ^{-2x }}{e ^ {2x }+ e ^{ -2 x }}

Given function  , Assume the    Now, back substituting the value of t,  , where C is any constant value.
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