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

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

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Key Concepts

Indefinite Integration

Indefinite integration involves finding a family of antiderivatives for a function where the result includes an arbitrary constant. Differences in the forms of antiderivatives, such as the ones shown, are acceptable as long as they differ only by a constant, since the derivative of any constant is zero.

Constant of Integration

The constant of integration represents the arbitrary constant in the general solution to an indefinite integral, accounting for all possible antiderivatives. When two integration results differ by an additive constant, they are still equivalent solutions.

Logarithmic Identities

Logarithmic identities allow the rewriting of logarithmic expressions in different but equivalent forms. For example, the relationship between logarithms, such as ln(sec(y)) and ln(sqrt(1+x^2)), can differ by a constant transformation, highlighting that these expressions can be made equivalent with proper manipulation.

Trigonometric Identities

Trigonometric identities and relationships are used to simplify and transform expressions involving inverse trigonometric functions and their corresponding trigonometric forms. Recognizing that functions like sec(arctan x) can be rewritten in terms of algebraic expressions such as sqrt(1+x^2) is key to understanding the equivalence of the two integration results.

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