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Magazine Chapter 6: Problem 20
Evaluate the integral. \(\int\left(\tan ^{2} x+\tan ^{4} x\right) d x\)
Short Answer
Expert verified
The integral evaluates to \( \frac{\tan^3 x}{3} + C \).
Step by step solution
01
Simplify the Integrand
We start with the integral \( \int (\tan^2 x + \tan^4 x) \, dx \). Notice that we can rewrite this as \( \int \tan^2 x (1 + \tan^2 x) \, dx \). This simplifies to \( \int \tan^2 x \sec^2 x \, dx \), using the identity \(1 + \tan^2 x = \sec^2 x\).
02
Use Substitution
Let \( u = \tan x \), then \( \frac{du}{dx} = \sec^2 x \) or \( du = \sec^2 x \, dx \). Substituting gives the integral \( \int u^2 \, du \).
03
Integrate Using the Power Rule
Apply the power rule for integration to \( \int u^2 \, du \). The power rule states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Thus, \( \int u^2 \, du = \frac{u^3}{3} + C \).
04
Substitute Back to Original Variable
We have \( \frac{u^3}{3} + C \) with \( u = \tan x \). Substitute back to get \( \frac{\tan^3 x}{3} + C \).
05
Express the Final Answer
Therefore, the integral \( \int (\tan^2 x + \tan^4 x) \, dx \) evaluates to \( \frac{\tan^3 x}{3} + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration techniques are methods used to find the integral of a function. Integrals can be challenging, but choosing the right technique simplifies the process. In this article, we focus on a few techniques relevant to the problem: substitution and simplifying the integrand.
- Substitution: This method is used when an integral contains a function and its derivative, like \(\int \tan^2 x \sec^2 x \, dx\). By substituting \(u = \tan x\), and knowing \(\frac{du}{dx} = \sec^2 x\), you can replace \(dx\) with \(du/\sec^2 x\). This converts the integral into a simpler form.
- Simplifying Integrands: Prerequisite to substitution is simplifying the integrand. For example, \(1 + \tan^2 x = \sec^2 x\) allows rewriting the integrand into a more manageable expression. This step often precedes substitution.
Trigonometric Substitution
Trigonometric substitution is a method used for integrating expressions containing trigonometric functions. It's particularly useful when dealing with integrals involving squares, roots, and complex trigonometric relationships.
- For the integral \(\int (\tan^2 x + \tan^4 x) \, dx\), we used the identity \(1 + \tan^2 x = \sec^2 x\) to simplify our task.
- Substituting identities transforms difficult expressions into simpler integrals, making them more manageable to solve.
Power Rule
The power rule is a straightforward technique in calculus for finding the integral of a polynomial function. It鈥檚 crucial in solving many basic integrals efficiently. For a function \( u^n \), the power rule states: \[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\] Here, \( C \) represents the constant of integration.
- This rule applies only when \( n eq -1 \).
- In the step-by-step solution, we applied it to \(\int u^2 \, du\) giving \(\frac{u^3}{3} + C\).
Definite and Indefinite Integrals
Integrals can be categorized into two types: definite and indefinite. Each serves a different purpose in calculus, providing either specific values or general expressions.
- Indefinite integrals: These are integrals with no specified limits. They provide a formula for the family of antiderivatives of a function. The answer always includes a constant of integration \( C \). In the exercise, the integral is indefinite, resulting in \(\frac{\tan^3 x}{3} + C\).
- Definite integrals: These are evaluated over a specific interval, yielding a specific number. There鈥檚 no constant \( C \) involved since the limits "subtract out" the constant effect.
