Question

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

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

Mathematical Methods in the Physical Sciences

Mathematical Methods in the Physical Sciences

Mary L. Boas 2nd Edition

Chapter 4, Problem 23

Instant Answer

Solved by Expert Raj Bala

01/21/2022

Step 1

Step 1: We are given the integral $\int_{0}^{\sin t} \frac{\sin ^{-1} x}{x} d x$ and we are asked to find its derivative with respect to $t$. Show more…

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

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Key Concepts

Fundamental Theorem of Calculus

This theorem establishes the relationship between differentiation and integration. It states that if a function is defined as an integral with a variable upper limit, then its derivative is given by the value of the integrand evaluated at that upper limit. This concept is crucial for converting problems about differentiating integrals into simpler evaluations of the original function.

Chain Rule

The chain rule is a fundamental tool for differentiating composite functions. When the upper limit of the integral is itself a function of another variable (such as sin t), the chain rule is applied to account for the derivative of this inner function. It ensures that all dependencies on the variable are correctly incorporated in the differentiation process.

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