### Problem Statement**Integrate using the trigonometric substitution method:**\[ \int \frac{3t - 9}{t^2 - 6t + 13} \, dt \]### ExplanationIn this problem, our goal is to evaluate the integral of the function $\frac{3t - 9}{t^2 - 6t + 13}$ with respect to $t$ by employing the trigonometric substitution method.This technique is often used for integrals involving algebraic expressions that can be transformed into trigonometric forms, making them easier to integrate. The expression $t^2 - 6t + 13$ in the denominator suggests a potential use for this method as it can be completed to form a perfect square.Key steps for solving integrals using trigonometric substitution typically involve:1. Completing the square in the quadratic expression.2. Choosing an appropriate trigonometric substitution based on the completed square form.3. Substituting and simplifying the integral.4. Solving the new integral in terms of the trigonometric variable.5. Converting back to the original variable $t$ upon integration.Each step needs careful consideration to ensure the integration process is correctly executed. It is essential to have a strong foundation in trigonometry and integration techniques to effectively solve this type of problem.

Answered: 3t – 9 Integrate using the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Concept explainers

Question

### Problem Statement
**Integrate using the trigonometric substitution method:**

\[ \int \frac{3t - 9}{t^2 - 6t + 13} \, dt \]

### Explanation

In this problem, our goal is to evaluate the integral of the function $\frac{3t - 9}{t^2 - 6t + 13}$ with respect to $t$ by employing the trigonometric substitution method.

This technique is often used for integrals involving algebraic expressions that can be transformed into trigonometric forms, making them easier to integrate. The expression $t^2 - 6t + 13$ in the denominator suggests a potential use for this method as it can be completed to form a perfect square.

Key steps for solving integrals using trigonometric substitution typically involve:
1. Completing the square in the quadratic expression.
2. Choosing an appropriate trigonometric substitution based on the completed square form.
3. Substituting and simplifying the integral.
4. Solving the new integral in terms of the trigonometric variable.
5. Converting back to the original variable $t$ upon integration.

Each step needs careful consideration to ensure the integration process is correctly executed. It is essential to have a strong foundation in trigonometry and integration techniques to effectively solve this type of problem.

Expert Solution

Step 1

Given:

$\int \frac{3 t - 9}{t^{2} - 6 t + 13} d t$

We have to integrate using the trigonometric substitution method.

Step 2

Consider,

$\int \frac{3 t - 9}{t^{2} - 6 t + 13} d t = \int \frac{3 (t - 3)}{t^{2} - 2 \times 3 t + 9 - 9 + 13} d t = \int \frac{3 (t - 3)}{{(t - 3)}^{2} + 4} d t [t^{2} - 2 \times 3 t + 9 = {(t - 3)}^{2}]$

Now Substitute, $t - 3 = 2 \tan (x) \Rightarrow d t = 2 s e c^{2} (x) d x$

We get,

$\int \frac{3 (t - 3)}{{(t - 3)}^{2} + 4} d t = \int \frac{3 (2 \tan (x))}{(2 \tan {(x))}^{2} + 4} 2 s e c^{2} (x) d x$

Step by step

Solved in 4 steps

Knowledge Booster

$Functions and Inverse Functions$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions