Special Integral Forms MCQ Quiz in मराठी - Objective Question with Answer for Special Integral Forms - मोफत PDF डाउनलोड करा

Last updated on Apr 14, 2025

पाईये Special Integral Forms उत्तरे आणि तपशीलवार उपायांसह एकाधिक निवड प्रश्न (MCQ क्विझ). हे मोफत डाउनलोड करा Special Integral Forms एमसीक्यू क्विझ पीडीएफ आणि बँकिंग, एसएससी, रेल्वे, यूपीएससी, स्टेट पीएससी यासारख्या तुमच्या आगामी परीक्षांची तयारी करा.

Latest Special Integral Forms MCQ Objective Questions

Top Special Integral Forms MCQ Objective Questions

Special Integral Forms Question 1:

Comprehension:

For the next two (2) items that follow:

The integral \(\smallint \frac{{{\rm{dx}}}}{{{\rm{acosx}} + {\rm{b}}\sin {\rm{x}}}}\) is of the form \(\frac{1}{{\rm{r}}}\ln \left[ {\tan \left( {\frac{{{\rm{x}} + {\rm{a}}}}{2}} \right)} \right]\)

What is r equal to?

  1. a2 + b2
  2. \(\sqrt {{{\rm{a}}^2} + {{\rm{b}}^2}} \)
  3. a + b
  4. \(\sqrt {{{\rm{a}}^2} - {{\rm{b}}^2}} \)

Answer (Detailed Solution Below)

Option 2 : \(\sqrt {{{\rm{a}}^2} + {{\rm{b}}^2}} \)

Special Integral Forms Question 1 Detailed Solution

Concept:

Sin (A + B) = sin (A) cos (B) + cos (A) sin (B)

\(\smallint {\rm{cosec\;xdx}} = - \ln \left| {{\rm{cosec\;x}} + \cot {\rm{x}}} \right| + {\rm{c}} = \ln \left| {\tan \left( {\frac{{\rm{x}}}{2}} \right)} \right| + {\rm{c}}b\)

 

Calculation:

\({\rm{I}} = \smallint \frac{{{\rm{dx}}}}{{{\rm{acosx}} + {\rm{b}}\sin {\rm{x}}}}\)

Let, a = r sin α and b = r cos α

\({\rm{I}} = \smallint \frac{{{\rm{dx}}}}{{{\rm{acosx}} + {\rm{b}}\sin {\rm{x}}}}\)

\( = \smallint \frac{{{\rm{dx}}}}{{{\rm{rsin\alpha cosx}} + {\rm{rcos\alpha }}\sin {\rm{x}}}}\)

\( = \smallint \frac{{{\rm{dx}}}}{{{\rm{r}}\left( {\sin \left( {{\rm{\alpha }} + {\rm{x}}} \right)} \right)}}\)

\( = \frac{1}{{\rm{r}}}\smallint {\rm{cosec\;}}\left( {{\rm{\alpha }} + {\rm{x}}} \right){\rm{dx}}\)

\( = \frac{1}{{\rm{r}}}\ln \left| {\tan \left( {\frac{{{\rm{\alpha }} + {\rm{x}}}}{2}} \right)} \right|\)

We have, a = r sin α and b = r cos α

Squaring and adding,

\({{\rm{a}}^2} + {{\rm{b}}^2} = {{\rm{r}}^2}\left( {{{\sin }^2}{\rm{\alpha }} + {{\cos }^2}{\rm{\alpha }}} \right)\)

\( \Rightarrow {\rm{r}} = \sqrt {{{\rm{a}}^2} + {{\rm{b}}^2}} \)

Hence, option (b) is correct.

Special Integral Forms Question 2:

Evaluate: \(\smallint \frac{{dx}}{{\sqrt {9 - 25{x^2}} }}\)

  1. \(\frac{1}{3} \cdot {\sin ^{ - 1}}\left( {\frac{{5x}}{3}} \right) + C\)
  2. \(\frac{1}{5} \cdot {\sin ^{ - 1}}\left( {\frac{{3x}}{5}} \right) + C\)
  3. \(\frac{1}{5} \cdot {\sin ^{ - 1}}\left( {\frac{{5x}}{3}} \right) + C\)
  4. \(\frac{1}{3} \cdot {\sin ^{ - 1}}\left( {\frac{{3x}}{5}} \right) + C\)

Answer (Detailed Solution Below)

Option 3 : \(\frac{1}{5} \cdot {\sin ^{ - 1}}\left( {\frac{{5x}}{3}} \right) + C\)

Special Integral Forms Question 2 Detailed Solution

Concept:

\(\smallint \frac{{dx}}{{\sqrt {{a^2} - {x^2}} }} = {\sin ^{ - 1}}\left( {\frac{x}{a}} \right) + C\)

Calculation:

Here we have to find the value of \(\smallint \frac{{dx}}{{\sqrt {9 - 25{x^2}} }}\)

The given integrand can be re-written as:

\(\Rightarrow \smallint \frac{{dx}}{{\sqrt {9 - 25{x^2}} }} = \frac{1}{5} \cdot \smallint \frac{{dx}}{{\sqrt {\frac{9}{{25}} - {x^2}} }}\)

\(= \frac{1}{5} \cdot \;\smallint \frac{{dx}}{{\sqrt {{{\left( {\frac{3}{5}} \right)}^2} - {x^2}} }}\)

Now by comparing,

 \(\smallint \frac{{dx}}{{\sqrt {{{\left( {\frac{3}{5}} \right)}^2} - {x^2}} }}\) with \(\smallint \frac{{dx}}{{\sqrt {{a^2} - {x^2}} }}\) we get, a = 3/5

As we know that,

 \(\smallint \frac{{dx}}{{\sqrt {{a^2} - {x^2}} }} = {\sin ^{ - 1}}\left( {\frac{x}{a}} \right) + C\) where C is a constant.

\(\Rightarrow \smallint \frac{{dx}}{{\sqrt {9 - 25{x^2}} }} = \frac{1}{5} \cdot {\sin ^{ - 1}}\left( {\frac{{5x}}{3}} \right) + C\) where C is a constant

Special Integral Forms Question 3:

Find the value of \(\rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+16}}\)

  1. \(\rm log\frac{ {|5+\sqrt {41}}|}{ {|4+\sqrt {32}}|}\)
  2. \(\rm log\frac{ {|5-\sqrt {41}}|}{ {|4+\sqrt {32}}|}\)
  3. \(\rm log\frac{ {|5+\sqrt {41}}|}{ {|4-\sqrt {32}}|}\)
  4. \(\rm log\frac{ {|5-\sqrt {41}}|}{ {|4-\sqrt {32}}|}\)
  5. None of these

Answer (Detailed Solution Below)

Option 1 : \(\rm log\frac{ {|5+\sqrt {41}}|}{ {|4+\sqrt {32}}|}\)

Special Integral Forms Question 3 Detailed Solution

Concept:

  • \(\rm \int\frac{dx}{\sqrt {x^{2}+a^{2}}}=log{\left |x+\sqrt {x^{2}+a^{2}}\right |}+C\)
  • \(\rm \int_{0}^{x}{f(x) dx}=F(x)-F(0)\), where F(x) is the anti-derivative of f(x).

Calculation:

Given: \(\rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+16}}\)

Using the formula, \(\rm \int\frac{dx}{\sqrt {x^{2}+a^{2}}}=log{\left |x+\sqrt {x^{2}+a^{2}}\right |}+C\)

\(\Rightarrow \rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+16}}=\left [ log{\left |x+\sqrt {x^{2}+16}\right |} \right ]_{4}^{5}\)

\(\Rightarrow \rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+16}}=log{|5+\sqrt {5^{2}+16}}|-log{|4+\sqrt {4^{2}+16}}|\)

\(=log{|5+\sqrt {41}}|-log{|4+\sqrt {32}}|\)

It is known that log a - log b = log (a / b)

\(\Rightarrow \rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}-16}}=log\frac{ {|5+\sqrt {41}}|}{ {|4+\sqrt {32}}|}\)

Hence, the correct answer is option 1.

Special Integral Forms Question 4:

Evaluate: \(\smallint \frac{{{x^2}}}{{\sqrt {{x^6} - 1} }}dx\)

  1. \(\frac{1}{4}\log \left| {{x^2} + \sqrt {{x^4} - 1} } \right| + C\)
  2. \(\frac{1}{3}\log \left| {{x^3} + \sqrt {{x^6} - 1} } \right| + C\)
  3. \(\frac{1}{3}\log \left| {{x^3} - \sqrt {{x^6} - 1} } \right| + C\)
  4. \(\frac{1}{4}\log \left| {{x^2} - \sqrt {{x^4} - 1} } \right| + C\)

Answer (Detailed Solution Below)

Option 2 : \(\frac{1}{3}\log \left| {{x^3} + \sqrt {{x^6} - 1} } \right| + C\)

Special Integral Forms Question 4 Detailed Solution

Concept:

  • \(\smallint \frac{{dx}}{{\sqrt {{x^2} - {a^2}} }} = \log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C\)

Calculation:

Here we have to find the value of \(\smallint \frac{{{x^2}}}{{\sqrt {{x^6} - 1} }}dx\)

Let x3 = t then 3x2 dx = dt

\(\Rightarrow \smallint \frac{{{x^2}}}{{\sqrt {{x^6} - 1} }}dx = \frac{1}{3}\;\smallint \frac{{dt}}{{\sqrt {{t^2} - 1} }}\)

Now by comparing \(\;\smallint \frac{{dt}}{{\sqrt {{t^2} - 1} }}\) with \(\smallint \frac{{dx}}{{\sqrt {{x^2} - {a^2}} }}\) we get, a = 1

As we know that, \(\smallint \frac{{dx}}{{\sqrt {{x^2} - {a^2}} }} = \log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C\) where C is a constant

\(\Rightarrow \frac{1}{3}\;\smallint \frac{{dt}}{{\sqrt {{t^2} - 1} }} = \frac{1}{3}\log \left| {t + \sqrt {{t^2} - 1} } \right| + C\)

Now by substituting x3 = t in the above equation we get,

\(\Rightarrow \smallint \frac{{{x^2}}}{{\sqrt {{x^6} - 1} }}dx = \frac{1}{3}\log \left| {{x^3} + \sqrt {{x^6} - 1} } \right| + C\) where C is a constant

Special Integral Forms Question 5:

Evaluate: \(\smallint \frac{{dx}}{{4 + 25{x^2}}}\)

  1. \(\frac{1}{{10}}{\tan ^{ - 1}}\left( {\frac{{5x}}{2}} \right) + C\)
  2. \(\frac{1}{{10}}{\tan ^{ - 1}}\left( {\frac{{2x}}{5}} \right) + C\)
  3. \(\frac{1}{{5}}{\tan ^{ - 1}}\left( {\frac{{5x}}{2}} \right) + C\)
  4. \(\frac{1}{{5}}{\tan ^{ - 1}}\left( {\frac{{2x}}{5}} \right) + C\)

Answer (Detailed Solution Below)

Option 1 : \(\frac{1}{{10}}{\tan ^{ - 1}}\left( {\frac{{5x}}{2}} \right) + C\)

Special Integral Forms Question 5 Detailed Solution

Concept:

  • \(\smallint \frac{{dx}}{{{x^2} + {a^2}}} = \frac{1}{a}{\tan ^{ - 1}}\frac{x}{a} + C\)

Calculation:

Here we have to find the value of \(\smallint \frac{{dx}}{{4 + 25{x^2}}}\)

The given integrand can be rewritten as:

\(\Rightarrow \smallint \frac{{dx}}{{4 + 25{x^2}}} = \frac{1}{{25}}\;\smallint \frac{{dx}}{{\left( {\frac{4}{{25}} + {x^2}} \right)}}\)

\(= \frac{1}{{25}}\;\smallint \frac{{dx}}{{\left\{ {{{\left( {\frac{2}{5}} \right)}^2} + {x^2}} \right\}}}\)

By comparing \(\;\smallint \frac{{dx}}{{\left\{ {{{\left( {\frac{2}{5}} \right)}^2} + {x^2}} \right\}}}\) with \(\smallint \frac{{dx}}{{{x^2} + {a^2}}}\) we get a = 2/5

Now as we know that, \(\smallint \frac{{dx}}{{{x^2} + {a^2}}} = \frac{1}{a}{\tan ^{ - 1}}\frac{x}{a} + C\) where C is a constant

\(\Rightarrow \smallint \frac{{dx}}{{4 + 25{x^2}}} = \frac{1}{{10}}{\tan ^{ - 1}}\left( {\frac{{5x}}{2}} \right) + C\)

Special Integral Forms Question 6:

What is \(\rm \int \frac{dx}{x(x^6+1)}\) equal to?

  1. \(\rm \frac{1}{6} \ln \left(\frac{x^6}{x^6 +1}\right)+c\)
  2. \(\rm \ln \left(\frac{x^6+1}{x^6}\right)+c\)
  3. \(\rm \ln \left(\frac{x^6}{x^6 +1}\right)+c\)
  4. \(\rm \frac{1}{6}\ln \left(\frac{x^6 +1}{x^6}\right)+c\)

Answer (Detailed Solution Below)

Option 1 : \(\rm \frac{1}{6} \ln \left(\frac{x^6}{x^6 +1}\right)+c\)

Special Integral Forms Question 6 Detailed Solution

Concept:

\(\rm \int \frac{1}{x^{2}-a^{2}}dx = \frac{1}{2a}log\left | \frac{x-a}{x+a} \right |\) 

Calculation:

I = \(\rm \int \frac{dx}{x(x^6+1)}\)

Let x6 + 1 = t               ....(i)

Differentiate both sides w.r.t x , 

\(\rm \frac{\mathrm{d} \left ( x^{6}+1 \right )}{\mathrm{d} x}= \frac{\mathrm{d} t}{\mathrm{d} x}\)

⇒ 6 x5 dx = dt

⇒ dx = \(\rm \frac{dt}{6x^{5}}\)            

Putting above values in given integration, 

I = \(\rm \frac{1}{6}\int \frac{1}{tx^{6}}dt\) 

I = \(\rm \frac{1}{6}\int \frac{1}{t(t-1)}dt\) 

I = \(\rm \frac{1}{6}\int \frac{1}{t^{2}-t}dt\) 

By compleating square Method , 

I = \(\rm \frac{1}{6}\int \frac{1}{t^{2}-t+\frac{1}{4}-\frac{1}{4}}dt\) 

I = \(\rm \frac{1}{6}\int \frac{1}{(t-1/2)^{2}-(1/2)^{2}}dt\)  

I = \(\rm \frac{1}{6}log\left | \frac{t-1}{t} \right |+C\)

From eq . ( i ) 

⇒ I = \(\rm \frac{1}{6} \ln \left(\frac{x^6}{x^6 +1}\right)+C\) .

The correct option is 1.

Special Integral Forms Question 7:

Comprehension:

For the next two (2) items that follow:

The integral \(\smallint \frac{{{\rm{dx}}}}{{{\rm{acosx}} + {\rm{b}}\sin {\rm{x}}}}\) is of the form \(\frac{1}{{\rm{r}}}\ln \left[ {\tan \left( {\frac{{{\rm{x}} + {\rm{a}}}}{2}} \right)} \right]\)

What is α equal to?

  1. \({\tan ^{ - 1}}\left( {\frac{{\rm{a}}}{{\rm{b}}}} \right)\)
  2. \({\tan ^{ - 1}}\left( {\frac{{\rm{b}}}{{\rm{a}}}} \right)\)
  3. \({\tan ^{ - 1}}\left( {\frac{{{\rm{a}} + {\rm{b}}}}{{{\rm{a}} - {\rm{b}}}}} \right)\)
  4. \({\tan ^{ - 1}}\left( {\frac{{{\rm{a}} - {\rm{b}}}}{{{\rm{a}} + {\rm{b}}}}} \right)\)

Answer (Detailed Solution Below)

Option 1 : \({\tan ^{ - 1}}\left( {\frac{{\rm{a}}}{{\rm{b}}}} \right)\)

Special Integral Forms Question 7 Detailed Solution

Calculation:

We have, a = r sin α and

b = r cos α

\(\therefore \frac{{\rm{a}}}{{\rm{b}}} = \frac{{{\rm{rsin\alpha }}}}{{{\rm{rcos\alpha }}}}\)

\(\Rightarrow \frac{{\rm{a}}}{{\rm{b}}} = \tan {\rm{\alpha }}\)

\( \Rightarrow {\rm{\alpha }} = {\tan ^{ - 1}}\left( {\frac{{\rm{a}}}{{\rm{b}}}} \right)\)

Hence, option (1) is correct.

Special Integral Forms Question 8:

Find a + b + c + d if \(\rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+9}}=log\frac{ {|a+\sqrt {b}}|}{ {|c+\sqrt {d}}|}\)

  1. 58
  2. 68
  3. 70
  4. 74

Answer (Detailed Solution Below)

Option 2 : 68

Special Integral Forms Question 8 Detailed Solution

Concept:

  • \(\rm \int\frac{dx}{\sqrt {x^{2}+a^{2}}}=log{\left |x+\sqrt {x^{2}+a^{2}}\right |}+C\)
  • \(\rm \int_{0}^{x}{f(x) dx}=F(x)-F(0)\), where F(x) is the anti-derivative of f(x).


Calculation:

Given: \(\rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+9}}=log\frac{ {|a+\sqrt {b}}|}{ {|c+\sqrt {d}}|}\)

Using the formula, \(\rm \int\frac{dx}{\sqrt {x^{2}+a^{2}}}=log{\left |x+\sqrt {x^{2}+a^{2}}\right |}+C\)

\(⇒ \rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+9}}= \left [ log{\left |x+\sqrt {x^{2}+9}\right |} \right ]_{4}^{5}\)

\(⇒ \rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+9}}=log{|5+\sqrt {5^{2}+9}}|-log{|4+\sqrt {4^{2}+9}}|=log{|5+\sqrt {34}}|-log{|4+\sqrt {25}}|\)

It is known that log a - log b = log a / b

\(⇒ \rm \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-16}}=log\frac{ {|5+\sqrt {34}}|}{ {|4+\sqrt {25}}|}\)--------(1)

∵ It is given that, \(\rm \int_{4}^{5} \frac{dx}{\sqrt {x^{2}+9}}=log\frac{ {|a+\sqrt {b}}|}{ {|c+\sqrt {d}}|}\)----------(2)

On comparing (1) and (2) we get, a = 5, b = 34, c = 4, d = 25

⇒ a + b + c+ d = 5 + 34 + 4 + 25 = 68

Hence, the correct answer is option 2.

Special Integral Forms Question 9:

Find the value of the \(\rm \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-16}}\)

  1. \(\rm log\frac{ {|6-\sqrt {20}}|}{{3}}\)
  2. \(\rm log\frac{ {|6+\sqrt {20}}|}{{5}}\)
  3. \(\rm log\frac{ {|3+\sqrt {20}}|}{{3}}\)
  4. \(\rm log\frac{ {|6+\sqrt {20}}|}{{8}}\)

Answer (Detailed Solution Below)

Option 4 : \(\rm log\frac{ {|6+\sqrt {20}}|}{{8}}\)

Special Integral Forms Question 9 Detailed Solution

Concept:

  • \(\rm \int\frac{dx}{\sqrt {x^{2}-a^{2}}}=log{\left |x+\sqrt {x^{2}-a^{2}}\right |}+C\)
  • \(\rm \int_{0}^{x}{f(x) dx}=F(x)-F(0)\), where F(x) is the anti-derivative of f(x).

Calculation:

Given: \(\rm \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-16}}\)

Using the formula, \(\rm \int\frac{dx}{\sqrt {x^{2}-a^{2}}}=log{\left |x+\sqrt {x^{2}-a^{2}}\right |}+C\)

\(\Rightarrow \rm \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-16}}= \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-4^{2}}}=\left [ log{\left |x+\sqrt {x^{2}-16}\right |} \right ]_{5}^{6}\)

Now, Substitute the limit to evaluate the value.

\(\Rightarrow \rm \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-16}}=log{|6+\sqrt {6^{2}-16}}|-log{|5+\sqrt {5^{2}-16}}|=log{|6+\sqrt {20}}|-log{|8}|\)

It is known that log a - log b = log (a / b)

\(\Rightarrow \rm \int_{5}^{6} \frac{dx}{\sqrt {x^{2}-16}}=log\frac{ {|6+\sqrt {20}}|}{{8}}\)

Hence, the correct answer is option 4.

Special Integral Forms Question 10:

Find the value of \(\rm \int_{0}^{1} \frac{dx}{\sqrt {1-x^{2}}}\)

  1. \(\rm sin^{-1}1\)
  2. \(\rm sin^{-1}(0.5)\)
  3. \(\rm sin^{-1}2\)
  4. \(\rm sin^{-1}0\)
    duplicate options found. English Question 1 options 1,2

Answer (Detailed Solution Below)

Option 1 : \(\rm sin^{-1}1\)

Special Integral Forms Question 10 Detailed Solution

Concept:

  • \(\rm \int \frac{dx}{\sqrt {a^{2}-x^{2}}}=sin^{-1}\frac{x}{a}+C\)
  • \(\rm \int_{0}^{x}{f(x) dx}=F(x)-F(0)\), where F(x) is the anti-derivative of f(x).


Calculation:

Given: \(\rm \int_{0}^{1} \frac{dx}{\sqrt {1^{2}-x^{2}}}\)

Using the formula, \(\rm \int \frac{dx}{\sqrt {a^{2}-x^{2}}}=sin^{-1}\frac{x}{a}+C\)

\(\rm \Rightarrow \int \frac{dx}{\sqrt {1-x^{2}}}=\int \frac{dx}{\sqrt {1^{2}-x^{2}}}=\left [ sin^{-1}x \right ]_{0}^{1}=sin^{-1}1-sin^{-1}0\)

It is known that, \(\rm sin^{-1}0=0\)

\(\rm \Rightarrow \int \frac{dx}{\sqrt {1-x^{2}}}=sin^{-1}1\)

Hence, the correct answer is option 1.

Get Free Access Now
Hot Links: teen patti master king teen patti rummy 51 bonus teen patti 100 bonus