Question

Consider the equation ${\int }_0^x(t^2-8t+13)dt=x\sin \left(\frac{a}{x}\right)$, then the number of real values of $x$ for which the equation has solution is,

• A. $1$
• B. $2$
• C. $3$
• D. infinite

Answer 1

The correct option is A $1$

After integrating above equation,

$\frac{x^3}{3}-4x^2+13x=x\sin \left(\frac{a}{x}\right)$

$\Rightarrow x^2-12x+39=3\sin \left(\frac{a}{x}\right)$, since $x=0$ is not in domain

$\Rightarrow (x-6{)}^2+3=3\sin (\frac{a}{x})$

Now the maximum possible value of RHS is $3$ and the minimum possible value of LHS is $3$

Therefore, $x=6$ is the only possible value

Hence, option 'A' is correct.