key term - ∫ tan^n x dx

Definition

The integral of the tangent function raised to the power of n, with respect to the variable x. This integral expression is commonly encountered in the context of trigonometric integrals, a fundamental concept in calculus.

5 Must Know Facts For Your Next Test

1. The integral of $\tan^n x$ dx can be evaluated using various techniques, such as trigonometric substitution or integration by parts.
2. The value of the integral $\int \tan^n x dx$ depends on the value of the exponent n, with different methods and results for different values of n.
3. When n is an even integer, the integral can be evaluated using trigonometric identities and integration by parts.
4. When n is an odd integer, the integral can be evaluated using trigonometric substitution, such as $u = \tan x$.
5. The evaluation of $\int \tan^n x dx$ is an important skill in solving problems related to trigonometric integrals and integration techniques.

Review Questions

• Explain the general approach to evaluating the integral $\int \tan^n x dx$, depending on the value of the exponent n.
  • The approach to evaluating the integral $\int \tan^n x dx$ depends on the value of the exponent n. When n is an even integer, the integral can be evaluated using trigonometric identities and integration by parts. For example, if n = 2, the integral can be rewritten as $\int \tan^2 x dx = \int (\sec^2 x - 1) dx = \tan x - x + C$. When n is an odd integer, the integral can be evaluated using trigonometric substitution, such as $u = \tan x$. This substitution simplifies the integral and allows for its evaluation using standard integration techniques.
• Describe the role of trigonometric substitution in evaluating the integral $\int \tan^n x dx$ when n is an odd integer.
  • When the exponent n in the integral $\int \tan^n x dx$ is an odd integer, trigonometric substitution can be used to simplify the integral and make it easier to evaluate. The typical substitution is $u = \tan x$, which means $du = \sec^2 x dx$. This substitution transforms the integral into a form that can be evaluated using standard integration techniques, such as power rule integration or integration by parts. The resulting expression in terms of the new variable u can then be converted back to the original variable x to obtain the final solution to the integral.
• Analyze the relationship between the evaluation of $\int \tan^n x dx$ and the concepts of trigonometric integrals and integration techniques.
  • The integral $\int \tan^n x dx$ is closely related to the broader topic of trigonometric integrals, which is a fundamental concept in calculus. The evaluation of this integral requires a deep understanding of trigonometric functions, identities, and integration techniques. Mastering the methods for solving $\int \tan^n x dx$, such as trigonometric substitution and integration by parts, demonstrates proficiency in applying various integration strategies to handle integrals involving trigonometric functions. Furthermore, the ability to evaluate $\int \tan^n x dx$ is crucial for solving more complex problems in areas like physics, engineering, and applied mathematics, where trigonometric integrals frequently arise.

Related terms

Trigonometric Functions:

Functions that describe the relationships between the sides and angles of right-angled triangles, such as sine, cosine, and tangent.

Trigonometric Substitution: A technique used to evaluate integrals involving trigonometric functions by substituting the trigonometric function with a new variable.

Integration by Parts:

A method of integration where the integral is broken down into two parts, one of which is integrated and the other is differentiated, to find the original integral.