The value of    \lim_{x\rightarrow 0}\frac{\int_{0}^{x^{2}}sec^{2}\, tdt}{x sin x}\; \; is

  • Option 1)

    2

  • Option 2)

    1  

  • Option 3)

    0

  • Option 4)

    3

 

Answers (1)

As we learnt in

NEWTON LEIBNITZ THEOREM -

\frac {d}{dt}\left ( \int_{f(t)}^{\phi (t))}F(x)dx \right )=F(\phi(t))\phi^{'}(t)-F(f(t))f^{'}(t)

-

 

 \lim_{x\rightarrow 0}\frac{\int_{0}^{x^{2}}\sec ^2tdt}{x\sin x}

Using L'Hospital rule and lebinitz rule

We get

\lim_{x\rightarrow 0}\frac{\sec ^2x^2 \times 2x-\sec^2 0 \times 0 }{\sin x+x\cos x}

\lim_{x\rightarrow 0}\frac{\sec ^2x^2 \times 2x}{\sin x+x\cos x }

\lim_{x\rightarrow 0}\frac{2\sec ^2x^2 }{\frac{sinx}{x}+x\cos x }

\Rightarrow \frac{2}{1+1}=1

 

 


Option 1)

2

This is incorrect option

Option 2)

1  

This is correct option

Option 3)

0

This is incorrect option

Option 4)

3

This is incorrect option

Preparation Products

JEE Main Rank Booster 2021

This course will help student to be better prepared and study in the right direction for JEE Main..

₹ 13999/- ₹ 9999/-
Buy Now
Knockout JEE Main April 2021 (Subscription)

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 4999/-
Buy Now
Knockout JEE Main April 2021

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 22999/- ₹ 14999/-
Buy Now
Knockout JEE Main April 2022

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 34999/- ₹ 24999/-
Buy Now
Knockout JEE Main January 2022

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 34999/- ₹ 24999/-
Buy Now
Exams
Articles
Questions