Question

$3-32$ Evaluate the integral. $$\int \arctan 4 t d t$$

   $3-32$ Evaluate the integral.
$$\int \arctan 4 t d t$$

Calculus Early Transcendentals

Calculus Early Transcendentals

James Stewart 6th Edition

Chapter 7, Problem 11

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Solved by Expert Tyler Moulton

08/13/2021

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We recognize that this is a problem that requires integration by parts. The formula for integration by parts is $\int u dv = uv - \int v du$. Show more…

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$3-32$ Evaluate the integral. $$\int \arctan 4 t d t$$

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Key Concepts

Integration by Parts

Integration by parts is a fundamental technique used to integrate products of functions. It is based on the product rule for differentiation and is given by the formula ?u dv = uv ? ?v du. This method is particularly useful when one function becomes simpler upon differentiation and the other is easily integrable.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, have specific derivatives that are often employed in integration problems. For example, the derivative of arctan(x) is 1/(1+x²), and understanding this relationship is important when integrating expressions involving inverse trigonometric functions.

Substitution Method

The substitution method, or change of variables, is used to simplify integrals by transforming them into a form that is easier to evaluate. In integrals where the argument of a function is a linear transformation (like 4t in arctan(4t)), substitution helps account for the effects of the chain rule in differentiation.

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