Question

Evaluate the integral. $\int \sin ^{-1} x d x$

   Evaluate the integral.
 $\int \sin ^{-1} x d x$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 10

Instant Answer

Solved by Expert Willis James

11/02/2024

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Step 1: Let's evaluate the integral $\int \sin^{-1} x \, dx$. Show more…

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Evaluate the integral. $\int \sin ^{-1} x d x$

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Key Concepts

Substitution Method

The substitution method involves changing variables to simplify an integral into a more manageable form. This technique is particularly useful when a direct antidifferentiation is complicated, and it is often used in conjunction with integration by parts for integrals involving inverse trigonometric functions.

Integration by Parts

Integration by parts is a technique used to integrate products of functions by choosing one function as u and the other as dv. It is based on the formula ? u dv = uv - ? v du, allowing one to simplify the integral by differentiating one function and integrating the other.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the arcsine function (sin?¹x), have unique derivative and integral properties that often appear in integration problems. Understanding these properties is essential for correctly differentiating and integrating expressions that involve these functions.

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