NIMCET 2011 — Mathematics PYQ

Step 1: Understand the Principle

The Fundamental Theorem of Calculus states that if $f(x)$ is defined by an integral of the form:

Then the derivative of $f(x)$ with respect to $x$ is simply the integrand evaluated at the upper limit:

Step 2: Apply the rule to the given function

In this problem, the integrand is $g(t)=t\sin t$.

The function is:

To find $f^{\prime }(x)$, we differentiate both sides with respect to $x$:

Step 3: Final Calculation

According to the rule, we simply replace the variable $t$ in the integrand with the upper limit $x$:

Conclusion:

The derivative of the given integral function is $x\sin x$.

Correct Option:

(b) $x\sin x$