Prove the following Definite Integration


$\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}}(\sin \sqrt{t}) d t}{x^{3}}$ is equal to:

  1. (1) $\frac{2}{3}$

  2. (2) 0

  3. (3) $\frac{1}{15}$

  4. (4) $\frac{3}{2}$

Correct Option: 1


$\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}$

$=\lim _{x \rightarrow 0} \frac{(\sin |x|) 2 x}{3 x^{2}}$

$=\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right) \times \frac{2}{3}$


Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now