Problem 23
Question
Find the indicated integral. $$\int \frac{\sec ^{2} t}{\tan t} d t$$
Step-by-Step Solution
Verified Answer
\( \int \frac{\sec ^{2} t}{\tan t} d t = \ln|\tan t| + C \)
1Step 1: Identify substitution
Recognize that the integrand is of the form \( \frac{f'(t)}{f(t)} \). Here, let \( u = \tan t \). Therefore, the derivative \( du = \sec^2 t \, dt \).
2Step 2: Substitution
Make the substitution \( u = \tan t \) and \( du = \sec^2 t \, dt \) in the integral \[ \int \frac{\sec^2 t}{\tan t} \, dt = \int \frac{1}{u} \, du. \]
3Step 3: Integrate
Recognize that \( \int \frac{1}{u} \, du \) is a standard integral. The result is: \[ \int \frac{1}{u} \, du = \ln|u| + C, \] where \( C \) is the constant of integration.
4Step 4: Substitute back
Replace \( u \) back with \( \tan t \): \[ \ln|u| + C = \ln|\tan t| + C. \]
Key Concepts
Integration by SubstitutionStandard IntegralsLogarithmic IntegralsDerivativesIntegration Techniques
Integration by Substitution
Integration by substitution helps simplify complex integrals by transforming them into more familiar forms.
In this exercise, we identified that the integrand \(\frac{\text{sec}^2 t}{\tan t}\) has a structure that can be simplified using substitution.
We let \(u = \tan(t)\), and thus the derivative \(du = \text{sec}^2 t \, dt\).
This substitution transformed the original integral \(\frac{\text{sec}^2 t}{\tan t} dt\) into a simpler form \(\frac{1}{u} du\), which is easier to integrate.
In this exercise, we identified that the integrand \(\frac{\text{sec}^2 t}{\tan t}\) has a structure that can be simplified using substitution.
We let \(u = \tan(t)\), and thus the derivative \(du = \text{sec}^2 t \, dt\).
This substitution transformed the original integral \(\frac{\text{sec}^2 t}{\tan t} dt\) into a simpler form \(\frac{1}{u} du\), which is easier to integrate.
Standard Integrals
Standard integrals are common integral forms that we can recognize and integrate directly.
In our problem, after applying substitution, the integral becomes \(\frac{1}{u} du\).
This is a standard integral and its solution is well-known:
\(\frac{1}{u} du\) integrates to \( \text{ln}|u| + C\).
Recognizing standard integrals is crucial for quickly solving integration problems.
In our problem, after applying substitution, the integral becomes \(\frac{1}{u} du\).
This is a standard integral and its solution is well-known:
\(\frac{1}{u} du\) integrates to \( \text{ln}|u| + C\).
Recognizing standard integrals is crucial for quickly solving integration problems.
Logarithmic Integrals
Logarithmic integrals are integrals that result in a natural logarithm function after integration.
In the exercise, after recognizing the standard integral form \(\frac{1}{u} du\), we derive the natural logarithm function:
\(\text{ln}|u| + C\).
Logarithmic integrals frequently appear in problems involving rational functions and require recognizing these patterns.
For our integral, this step was straightforward, resulting in the logarithm of the absolute value of \(u\).
In the exercise, after recognizing the standard integral form \(\frac{1}{u} du\), we derive the natural logarithm function:
\(\text{ln}|u| + C\).
Logarithmic integrals frequently appear in problems involving rational functions and require recognizing these patterns.
For our integral, this step was straightforward, resulting in the logarithm of the absolute value of \(u\).
Derivatives
Derivatives are crucial in identifying the correct substitutions in integration.
For this problem, we identified \(u = \tan(t)\) and used its derivative \(du = \text{sec}^2 t \, dt\).
Recognizing which function to use for substitution often relies on understanding how derivatives and integrals relate.
Knowing common derivatives allows us to simplify integrals more effectively.
For this problem, we identified \(u = \tan(t)\) and used its derivative \(du = \text{sec}^2 t \, dt\).
Recognizing which function to use for substitution often relies on understanding how derivatives and integrals relate.
Knowing common derivatives allows us to simplify integrals more effectively.
Integration Techniques
Various integration techniques help solve different types of integrals.
In this exercise, we used substitution, a powerful method to simplify complex integrals.
Other techniques include integration by parts, partial fractions, and trigonometric identities.
Choosing the right technique often depends on the form of the integrand.
Substitution was ideal here due to the form of the integrand \(\frac{\text{sec}^2 t}{\tan t}\).
Mastering different techniques enhances your ability to tackle a diverse range of integral problems.
In this exercise, we used substitution, a powerful method to simplify complex integrals.
Other techniques include integration by parts, partial fractions, and trigonometric identities.
Choosing the right technique often depends on the form of the integrand.
Substitution was ideal here due to the form of the integrand \(\frac{\text{sec}^2 t}{\tan t}\).
Mastering different techniques enhances your ability to tackle a diverse range of integral problems.
Other exercises in this chapter
Problem 21
Find the indicated integral. $$\int \sin x \cos x d x$$
View solution Problem 22
Find the indicated integral. $$\int x \sin x d x$$
View solution Problem 25
Find the indicated integral. $$\int_{0}^{\pi} \cos \left(\frac{x}{3}\right) d x$$
View solution Problem 26
Find the indicated integral. $$\int_{0}^{1} x \sin \left(x^{2}\right) d x$$
View solution