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Chapter 4: Problem 23

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

01

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
02

Differentiate the Upper Limit

Substitute the upper limit of our integral, and
03

Evaluate the Integral

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

If \(u=x^{2}\left(x^{2}+y^{2}\right)\), find \(\partial u / \partial x, \partial u / \partial y\).

Use L'Hôpital's rule to evaluate \(\lim _{x \rightarrow 2} \frac{1}{x-2} \int_{2}^{x} \frac{\sin t}{t} d t\).

Find the two-variable Maclaurin series for the following functions. \(\cos x \sinh y\)

The temperature at a point \((x, y, z)\) in the sphere \(x^{2}+y^{2}+z^{2} \leq 1\) is given by \(T=y^{2}+x z\). Find the largest and smallest values which \(T\) takes (a) on the circle \(y=0, x^{2}+z^{2}=1\), (b) on the surface \(x^{2}+y^{2}+z^{2}=1\), (c) in the whole sphere.

For \(u=e^{x} \cos y\), (a) verify that \(\partial^{2} u / \partial x \partial y=\partial^{2} u / \partial y \partial x\); (b) verify that \(\partial^{2} u / \partial x^{2}+\partial^{2} u / \partial y^{2}=0\).
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