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

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

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

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Antiderivatives and the Constant of Integration

When finding an antiderivative, the result includes an arbitrary constant because differentiation of a constant yields zero. This means that two different-looking answers can be equivalent if they differ only by a constant value. In the context of the problem, both integration formulas are valid since their difference is merely a constant, consistent with the fundamental theorem of calculus.

Equivalence of Different Integration Methods

Different integration techniques may lead to distinct expressions for the same antiderivative. Although these expressions may appear different at first glance, they can be algebraically manipulated to show they differ only by a constant. This key insight confirms that both methods yield correct results as long as their derivatives match the original integrand.

Logarithmic and Trigonometric Relationships

The integration of inverse trigonometric functions often involves logarithmic expressions and trigonometric identities. Understanding how expressions like ln(sec(arctan x)) compare to ln(sqrt(1+x^2)) is crucial, because these relationships are derived from the identities involving the trigonometric and inverse trigonometric functions. Recognizing these connections allows one to verify the consistency of different antiderivative forms.

Transcript

Please give Ace some feedback