Compare the results of using Equations (4) and (5)....

Compare the results of using Equations (4) and (5). Equations (4) and (5) lead to different formulas for the integral of $\tan ^{-1} x$ a. $\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sec \left(\tan ^{-1} x\right)+C$ Eq. (4) b. $\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sqrt{1+x^{2}}+C$ Eq. (5) Can both integrations be correct? Explain.

Compare the results of using Equations (4) and (5).
Equations (4) and (5) lead to different formulas for the integral of $\tan ^{-1} x$
a. $\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sec \left(\tan ^{-1} x\right)+C$
Eq. (4)
b. $\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sqrt{1+x^{2}}+C$
Eq. (5)
Can both integrations be correct? Explain.

Key Concepts

Antiderivatives and the Constant of Integration

In indefinite integrals any two antiderivatives of the same function can differ only by an arbitrary constant. This means that even if two results appear different at first glance, they are equivalent as long as their difference is merely an additive constant.

Integration by Parts

Integration by parts is a fundamental technique used to integrate products of functions. When integrating functions involving inverse trigonometric terms, this method is particularly useful and can lead to different looking but equivalent antiderivatives.

Trigonometric Identities Involving Inverse Functions

Inverse trigonometric functions have inherent relationships with other trigonometric functions. By applying these identities—such as expressing a trigonometric function of an inverse trigonometric expression in terms of algebraic functions—one can simplify and verify the equivalence of seemingly different integral expressions.

Equivalence of Different Forms of Antiderivatives

Different integration methods may yield antiderivatives that appear distinct due to variations in algebraic manipulations and the presentation of logarithmic or trigonometric expressions. However, these forms represent the same family of functions if they differ only by a constant, highlighting the flexibility in the integration process.

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