Mathematics
ASSERTION

If $$\int \sec \left ( \log x \right )\left ( 1+\tan \left ( \log x \right ) \right )dx=f\left ( x \right )+C$$ then $$f\left ( x \right )=x\sec \left ( \log x \right )$$

REASON

$$\int \left \{ x{f}'\left ( x \right )+f\left ( x \right ) \right \}dx=xf\left ( x \right )+C$$


ANSWER

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion


SOLUTION
Given $$\int \sec \left ( \log x \right )\left ( 1+\tan \left ( \log x \right ) \right )dx=f\left ( x \right )+C$$        ....(1)
Consider, $$I= \int \sec \left ( \log x \right )\left ( 1+\tan \left ( \log x \right ) \right )dx$$
Putting $$\log x=t$$
$$dx=e^{t}dt$$
$$I=\int e^{t}\left ( \sec t+\sec t\tan t \right )dt$$
   $$=e^{t}\sec t+C$$              ($$\because \int e^x(f(x)+f'(x))=e^f(x)+C$$)
   $$=x\sec \left ( \log x \right )+C$$
So, by (1), we get
$$f\left ( x \right )=x\sec \left ( \log x \right )$$
$$\therefore $$   Assertion (A) is true.

Reason(R): $$\int \left \{ f\left ( x \right )+x{f}'\left ( x \right ) \right \}dx$$
     $$=\int f\left ( x \right )dx+\int x{f}'\left ( x \right )dx$$
     $$=f\left ( x \right )+xf\left ( x \right )-\int 1.f\left ( x \right )dx$$
     $$=xf\left ( x \right )+C$$
$$\therefore $$   Reason (R) is also true but not the proper explanation for Assertion (A).
View Full Answer

Its FREE, you're just one step away


Assertion & Reason Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Integrate :
$$\int {\cfrac{{6\sin \left( x \right)co{s^2}\left( x \right) + sin\left( {2x} \right) - 23\sin \left( x \right)}}{{{{\left( {\cos \left( x \right) - 1} \right)}^2}\left( {5 - {{\sin }^2}\left( x \right)} \right)}}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 TRUE/FALSE Medium
State true or false:
The value of the integral $$\displaystyle \int_{-\infty }^{0}x.e^{x}dx$$ is not finite.
  • A. True
  • B. False

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
$$\int\limits_0^\infty  {\dfrac{{{x^2} + 1}}{{{x^4} + 10{x^2} + 9}}\;{\text{dx}}\;{\text{ = }}} $$
  • A. $$\pi $$
  • B. $$\dfrac{\pi }{2}$$
  • C. $$\dfrac{\pi }{3}$$
  • D. $$\dfrac{\pi }{6}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard
Evaluate the given integral.
$$\displaystyle\int { \cfrac { { x }^{ 9 } }{ { \left( 4{ x }^{ 2 }+1 \right)  }^{ 6 } }  } dx $$ 
  • A. $$\cfrac { 1 }{ 5x } { \left( 4+\cfrac { 1 }{ { x }^{ 2 } } \right) }^{ -5 }+C$$
  • B. $$\cfrac { 1 }{ 5 } { \left( 4+\cfrac { 1 }{ { x }^{ 2 } } \right) }^{ -5 }+C$$
  • C. $$\cfrac { 1 }{ 10x } { \left( \cfrac { 1 }{ { x }^{ 2 } } +4 \right) }^{ -5 }+C$$
  • D. $$\cfrac { 1 }{ 10 } { \left( \cfrac { 1 }{ { x }^{ 2 } } +4 \right) }^{ -5 }+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Solve $$\displaystyle\int \dfrac{x}{{{{\left( {x + 1} \right)}^2}}}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer