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

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Identify the integral to solve: \(\int \tan^{-1}(\sqrt{y}) \, dy\).

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

Differentiate \(y = u^2\) with respect to \(u\) to find \(dy\): \(dy = 2u \, du\).

Rewrite the integral in terms of \(u\): \(\int \tan^{-1}(u) \, dy = \int \tan^{-1}(u) \cdot 2u \, du\).

Now, the integral is \(2 \int u \tan^{-1}(u) \, du\), which can be approached using integration by parts or found in an integral table.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 arctan (tan⁻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 helps in applying substitution and integration techniques.

Using Integral Tables

Integral tables provide formulas for standard integrals, allowing quick evaluation once the integral is transformed appropriately. After substitution, matching the integral to a form in the table simplifies the process. Familiarity with common integral forms and how to use these tables is crucial for efficient problem-solving.

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