Question

lf $f\left(x\right)={\int }_1^x\frac{z^4\sin \frac{1}{z}+z^2}{1+|z{|}^3}dz$ then $\left(f^{\prime }\right(x\left)\right)$ at $x=1$ equals

• A. $1+\sin 1$
• B. $2$
• C. $\frac{1}{2}(\frac{\pi }{2}+1)$
• D. $\frac{1+\sin 1}{2}$

Answer 1

Answer: (D)

$\frac{1+\sin 1}{2}$

$f\left(x\right)={\int }_1^x\frac{z^4\sin \frac{1}{z}+z^2}{1+{\left|z\right|}^3}dz$
$f^{\prime }\left(x\right)=\frac{1.x^4\sin \frac{1}{x}+x^2}{1+{\left|x\right|}^3}$ ...... Using Leibniz Rule
At $x=1$, we get
$f^{\prime }\left(1\right)=\frac{\sin 1+1}{2}$

Hence, option D is correct.