Question

Evaluate the integral. $$ \int \tan ^{6} x d x $$

   Evaluate the integral.
$$
\int \tan ^{6} x d x
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 9, Problem 13

Instant Answer

Solved by Expert Nick Johnson

03/20/2022

Step 1

Step 1: First, we rewrite the integral of $\tan^6x$ as the integral of $\tan^4x$ times $\tan^2x$: $$ \int \tan ^{6} x d x = \int \tan^4x (\tan^2x) dx $$ Show more…

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Evaluate the integral. $$ \int \tan ^{6} x d x $$

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Key Concepts

Trigonometric Identities

Trigonometric identities, such as tan^2(x) = sec^2(x) - 1, are essential when working with powers of trigonometric functions. These identities allow you to rewrite and simplify expressions, reducing the complexity of the integral and transforming it into a form that is more amenable to standard integration techniques.

Power Reduction Techniques

When integrating high powers of trigonometric functions, power reduction techniques help to break them down into lower powers. This involves expressing even powers of tangent in terms of secant functions or using identities to reduce the degree of the power, thereby facilitating the use of basic integration formulas.

Integration Techniques for Trigonometric Functions

Integrating trigonometric functions often requires a combination of techniques such as substitution, algebraic manipulation, and splitting the integrals into simpler parts. These techniques are particularly useful for functions raised to a power, like tan^6(x), which might be expressed in a form that can be integrated term-by-term.

Substitution Method

The substitution method is a key integration technique where a new variable is introduced to simplify the integral. For trigonometric integrals involving tan(x), using the substitution u = tan(x) (which implies that du = sec^2(x) dx) can streamline the process by converting the integral into a polynomial form in terms of u that is often easier to handle.

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