Problem 19
Question
Perform the indicated integrations. $$ \int \tan ^{4} x d x $$
Step-by-Step Solution
Verified Answer
\( \int \tan^4 x \, dx = \tan x - \frac{2}{3} \tan^3 x + x + C \).
1Step 1: Express the function in simpler terms
The given integral is \( \int \tan^4 x \, dx \). We'll start by expressing \( \tan^4 x \) in simpler terms using the identity \( \tan^2 x = \sec^2 x - 1 \). So we rewrite \( \tan^4 x \) as \( (\tan^2 x)^2 = (\sec^2 x - 1)^2 \).
2Step 2: Expand the expression
Now that we have \( (\sec^2 x - 1)^2 \), we expand this expression: \((\sec^2 x - 1)^2 = \sec^4 x - 2\sec^2 x + 1 \). Thus, the integral becomes \( \int(\sec^4 x - 2\sec^2 x + 1) \, dx \).
3Step 3: Integrate term by term
We now integrate each term separately: 1. \( \int \sec^4 x \, dx \)2. \( -2\int \sec^2 x \, dx \)3. \( \int 1 \, dx \).
4Step 4: Solve \(\int \sec^4 x \, dx\)
We can integrate \(\sec^4 x\) using the reduction formula: \(\int \sec^4 x \, dx = \int (\sec^2 x)^2 \, dx = \int (1 + \tan^2 x)\sec^2 x \, dx = \int \sec^2 x \, dx + \int \tan^2 x\sec^2 x \, dx\). By solving, we find: \(\int \sec^2 x \, dx = \tan x + C \, \text{and} \, \int \tan^2 x \sec^2 x \, dx = \frac{1}{3} \tan^3 x + C.\)
5Step 5: Solve \(\int \sec^2 x \, dx \) and \(\int 1 \, dx \)
The integral \( -2\int \sec^2 x \, dx = -2 \tan x + C \).For \( \int 1 \, dx \), the integral is simply \( x + C \).
6Step 6: Combine the results
Now, combine the results from each part:\[ \int \tan^4 x \, dx = \int \sec^4 x \, dx - 2\int \sec^2 x \, dx + \int 1 \, dx \]\[ = (\tan x + \frac{1}{3}\tan^3 x + C) - 2\tan x + x + C \]Simplify to get \( \tan x - \frac{2}{3}\tan^3 x + x + C \).
Key Concepts
Trigonometric functionsIntegration techniquesReduction formula
Trigonometric functions
Trigonometric functions are fundamental in studying angles and solving problems in calculus. They include functions like sine, cosine, and tangent, each with unique properties and formulas that are essential for integration. These functions are defined based on relationships between angles and lengths in right triangles.
For example, the tangent function, denoted as \( \tan x \), is defined as the ratio of the sine and cosine of \( x \), i.e., \( \tan x = \frac{\sin x}{\cos x} \).
This means that tangent can be expressed in terms of sine and cosine, which is useful when integrating expressions involving tangent. Understanding these relationships and identities, such as \( \tan^2 x = \sec^2 x - 1 \), helps simplify complex trigonometric expressions, making them easier to integrate.
For example, the tangent function, denoted as \( \tan x \), is defined as the ratio of the sine and cosine of \( x \), i.e., \( \tan x = \frac{\sin x}{\cos x} \).
This means that tangent can be expressed in terms of sine and cosine, which is useful when integrating expressions involving tangent. Understanding these relationships and identities, such as \( \tan^2 x = \sec^2 x - 1 \), helps simplify complex trigonometric expressions, making them easier to integrate.
Integration techniques
Integration techniques are methods used to solve integrals, especially when dealing with complex functions. There are several techniques that can be applied depending on the form of the integral. In our task of integrating \( \int \tan^4 x \, dx \), multiple techniques are employed to simplify the problem.
- **Substitution:** Often used when an integral contains a function and its derivative. This method changes variables to transform the integrand into a more manageable form.
- **Integration by Parts:** Utilizes the product rule for differentiation and is helpful for integrals that are products of two functions.
- **Trigonometric Identities:** Key identities such as \( \tan^2 x = \sec^2 x - 1 \) alter trigonometric expressions, simplifying the integration process.
Reduction formula
The reduction formula is a powerful tool in calculus for simplifying repetitive integrals. It expresses an integral in terms of another integral with a lower power or simpler form. This technique is particularly helpful for dealing with powers of trigonometric functions.
For instance, to integrate \( \sec^4 x \, dx \), we can use the reduction formula to break it down as follows:\[ \int \sec^4 x \, dx = \int (1 + \tan^2 x)\sec^2 x \, dx = \int \sec^2 x \, dx + \int \tan^2 x \sec^2 x \, dx \].
This turns a complex higher power integration into a sum of simpler integrals, each of which can be calculated using fundamental techniques.
The reduction formula simplifies the task by methodically reducing the power of the integrand until it becomes a basic integral, as seen in the example where \( \tan^4 x \) is integrated step-by-step using previously solved simpler integrals.
For instance, to integrate \( \sec^4 x \, dx \), we can use the reduction formula to break it down as follows:\[ \int \sec^4 x \, dx = \int (1 + \tan^2 x)\sec^2 x \, dx = \int \sec^2 x \, dx + \int \tan^2 x \sec^2 x \, dx \].
This turns a complex higher power integration into a sum of simpler integrals, each of which can be calculated using fundamental techniques.
The reduction formula simplifies the task by methodically reducing the power of the integrand until it becomes a basic integral, as seen in the example where \( \tan^4 x \) is integrated step-by-step using previously solved simpler integrals.
Other exercises in this chapter
Problem 19
Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. \(\int \frac{3 x}{\sqrt{x^{2}+2 x+5}} d x
View solution Problem 19
Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{4}+8 x^{2}+8}{x^{3}-4 x} d x\)
View solution Problem 19
Perform the indicated integrations. $$ \int \frac{\sin \left(\ln 4 x^{2}\right)}{x} d x $$
View solution Problem 19
$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int z^{3} \ln z d z $$
View solution