Question

Evaluate the integral. $\int_0^{1 / 2} \cos ^{-1} x d x$

   Evaluate the integral.
$\int_0^{1 / 2} \cos ^{-1} x d x$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 31

Instant Answer

Solved by Expert Linda Hand

02/15/2022

Step 1

Let's use integration by parts with the formula $\int u \, dv = uv - \int v \, du$. Show more…

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Evaluate the integral. $\int_0^{1 / 2} \cos ^{-1} x d x$

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Key Concepts

Integration by Parts

Integration by parts is a method used to integrate products of functions by isolating one function to differentiate and another to integrate. It is based on the product rule of differentiation and is particularly useful when dealing with functions that are not straightforward to integrate directly, such as those involving inverse trigonometric functions. The formula for integration by parts is ? u dv = uv ? ? v du.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the arccosine (or cos?¹) function, are functions that reverse the operation of the trigonometric functions. They are defined for specific domains and are useful for retrieving angles from given trigonometric ratios. Understanding these functions includes knowing their ranges, derivatives, and how they are used in various contexts in calculus.

Definite Integration

Definite integration involves computing the antiderivative of a function and then evaluating it at the endpoints of the integration interval. This process gives the net area under the curve of the function over that interval. Familiarity with definite integrals includes understanding the Fundamental Theorem of Calculus, which connects antiderivatives with the evaluation of areas.

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