Question

Find \(\frac{d}{d t} \int_{0}^{\sin t} \frac{\sin ^{-1} x}{x} d x\).

Short answer

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Step-by-step solution

Apply the Leibniz Rule

The Leibniz rule for differentiating an integral with variable limits is Subtract-and-add give us where we assume and where and

Differentiate the Upper Limit

Substitute the upper limit of our integral, and

Evaluate the Integral

Now evaluate the function using the chain and product rule as necessary to obtain where

Key concepts

Leibniz Rule

The Leibniz rule is a powerful tool for differentiating integrals that have variable upper and lower limits. When an integral \(\frac{d}{d t} \int_{a(t)}^{b(t)} f(x,t) d x\) has both of its limits as functions of a variable, the Leibniz rule helps to solve it. According to this rule: \[ \frac{d}{d t} \int_{a(t)}^{b(t)} f(x,t) dx = f(b(t),t) b'(t) - f(a(t),t) a'(t) + \int_{a(t)}^{b(t)} \frac{\partial }{\partial t}f(x,t) dx \] This means we differentiate with respect to the upper limit, lower limit, and the integrand. Remember these steps when you see variable limits in an integral.

Integral Differentiation

Integral differentiation involves finding the derivative of an integral. Here, the exercise requires us to find \(\frac{d}{d t} \int_{0}^{\sin t} \frac{\sin^{-1} x}{x} dx\). First, identify our function \(f\) as \(\frac{\sin^{-1} x}{x} \). Next, we apply the chain rule separately to each variable limit. Plug in \( \sin t \) at the upper limit and 0 for the lower limit. Use the Leibniz rule for variable limits, It then simplifies by taking the derivative of the inside function. With practice, this process will become intuitive.

Variable Limits

Variable limits mean the limits of integration are not constants, but functions of a variable. To differentiate an integral like \(\frac{d}{d t} \int_{0}^{\sin t} \frac{\sin^{-1} x}{x} dx\), you follow a methodical process:

• Identify the upper \(b(t)\) and lower \(a(t)\) limits of the integral.
• Use the Leibniz rule to structure your solution.
• Evaluate each term carefully to find the derivative of the integral.
• Observe how changes in limits affect your result.

Understanding these steps thoroughly helps you apply the technique accurately.