Equations (4) and (5) lead to different formulas for the...

Equations $(4)$ and $(5)$ lead to different formulas for the integral of $\tan ^{-1} x :$
a. $\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sec \left(\tan ^{-1} x\right)+C$
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

Integration by Parts

Integration by parts is a fundamental technique in calculus used to integrate products of functions. It is often applied when directly integrating an expression is challenging. In the context of this problem, integration by parts is applied to the inverse tangent function, leading to two different-looking antiderivatives that are mathematically equivalent up to a constant.

Antiderivative Uniqueness (Constant of Integration)

In indefinite integration, antiderivatives are unique only up to an additive constant. This means that two expressions for an antiderivative can look different but still represent the same family of functions if they differ by a constant value. This principle explains why the two formulas obtained for the integral of arctan x are both correct.

Trigonometric Identities and Inverse Functions

Trigonometric identities can be used to simplify expressions involving inverse trigonometric functions. For example, knowing that sec(arctan x) simplifies to ?(1+x²) shows that the logarithmic terms in the two formulas are identical except for their form. This identity is crucial to understanding how the two integration results are equivalent despite their different appearances.

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