Question

Evaluate the integral. $\int \arctan 4 t d t$

   Evaluate the integral.
$\int \arctan 4 t d t$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 11

Instant Answer

Solved by Expert Tyler Moulton

08/13/2021

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The formula for integration by parts is \(\int u \, dv = uv - \int v \, du\). Show more…

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Evaluate the integral. $\int \arctan 4 t d t$

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Key Concepts

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and allows one to express the integral of a product in terms of the integral of a simpler product. This method is particularly useful when one of the functions in the product has a simpler derivative, as is often the case with inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to reverse the trigonometric relationships and are central in many areas of calculus. Their unique derivatives and antiderivatives often require specialized integration techniques, and understanding their properties is key to correctly handling integrals that involve them.

Substitution Method

The substitution method, also known as u-substitution, is a technique used to simplify integrals by changing variables. This is particularly useful when the integral involves a composite function, as it allows the integrand to be rewritten in a form that matches a standard integral. It plays a supportive role when dealing with integrals of functions like arctan, especially if a constant multiple within the argument complicates the expression.

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