In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ cos^(-1)(√x) / √x dx

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Identify the integral: \(\int \frac{\cos^{-1}(\sqrt{x})}{\sqrt{x}} \, dx\).

Choose a substitution to simplify the integral. Let \(u = \sqrt{x}\), which implies \(x = u^2\).

Differentiate \(x = u^2\) to find \(dx\): \(dx = 2u \, du\).

Rewrite the integral in terms of \(u\): replace \(\sqrt{x}\) with \(u\) and \(dx\) with \(2u \, du\). The integral becomes \(\int \frac{\cos^{-1}(u)}{u} \cdot 2u \, du\).

Simplify the integral expression: the \(u\) terms cancel, leaving \(2 \int \cos^{-1}(u) \, du\). Now, use integration techniques or a table of integrals to evaluate \(\int \cos^{-1}(u) \, du\).

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주요 개념

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves choosing a substitution that transforms the integral into a more familiar or standard form, making it easier to evaluate. This technique is especially useful when the integral contains composite functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccosine (cos⁻¹), are the inverses of the basic trigonometric functions. Understanding their properties and derivatives is essential for integrating expressions involving them. Recognizing how to handle these functions within integrals helps in applying substitution effectively.

Using Integral Tables

Integral tables provide standard forms of integrals and their solutions, which can save time in evaluation. After substitution simplifies the integral, matching it to a form in the table allows direct application of known results. Familiarity with common integral forms is key to using these tables efficiently.

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ cos^(-1)(√x) / √x dx