Question

Perform the indicated integrations. $$\int \tan ^{3} x d x$$

   Perform the indicated integrations.
$$\int \tan ^{3} x d x$$

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 7, Problem 21

Instant Answer

Solved by Expert Rukhmani Jain

06/13/2021

Step 1

Step 1: First, we rewrite the integral as a product of $\tan x$ and $\tan^2 x$: $$\int \tan ^{3} x d x = \int \tan x \cdot \tan^2 x dx$$ Show more…

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Perform the indicated integrations. $$\int \tan ^{3} x d x$$

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Key Concepts

Trigonometric Identities

These are formulas that relate the various trigonometric functions. For instance, rewriting tan^2 x as sec^2 x - 1 is crucial when dealing with powers of tangent, as it allows the integral to be split into simpler parts that are easier to integrate.

Substitution Method

A technique where one replaces a complicated expression with a single variable, simplifying the integration process. In problems involving tan^3 x, substitution is often used because the derivative of tan x is sec^2 x, which naturally appears when expressing tan^3 x in a more manageable form.

Integration of Trigonometric Functions

This encompasses the methods and standard integrals related to sine, cosine, tangent, and their combinations. It includes knowing how to integrate functions like tan x and sec^2 x, and applying these results when dealing with powers or products of trigonometric functions.

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