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...
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Question
![**Trigonometric Identity and Integral Evaluation**
Use a trigonometric identity to evaluate the integral.
\[
\int \tan^2(x) \, dx = \underline{\hspace{2cm}} + C
\]
**Explanation:**
This problem asks you to find the integral of \(\tan^2(x)\) using a trigonometric identity. To solve this:
1. Recall that \(\tan^2(x) = \sec^2(x) - 1\).
2. Substitute this identity into the integral:
\[
\int \tan^2(x) \, dx = \int (\sec^2(x) - 1) \, dx
\]
3. Integrate term by term:
\[
\int \sec^2(x) \, dx - \int 1 \, dx = \int \sec^2(x) \, dx - x
\]
4. Recall that \(\int \sec^2(x) \, dx = \tan(x)\).
5. Therefore, the integral evaluates to:
\[
\tan(x) - x + C
\]
**Note:** Don't forget to add the constant of integration \(C\) at the end.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce84b589-8243-4834-a602-7a7ad7e8662d%2F290c0e3a-5a54-4ebc-b401-11a4131dc5f0%2Fvx92iol_processed.png&w=3840&q=75)
Transcribed Image Text:**Trigonometric Identity and Integral Evaluation**
Use a trigonometric identity to evaluate the integral.
\[
\int \tan^2(x) \, dx = \underline{\hspace{2cm}} + C
\]
**Explanation:**
This problem asks you to find the integral of \(\tan^2(x)\) using a trigonometric identity. To solve this:
1. Recall that \(\tan^2(x) = \sec^2(x) - 1\).
2. Substitute this identity into the integral:
\[
\int \tan^2(x) \, dx = \int (\sec^2(x) - 1) \, dx
\]
3. Integrate term by term:
\[
\int \sec^2(x) \, dx - \int 1 \, dx = \int \sec^2(x) \, dx - x
\]
4. Recall that \(\int \sec^2(x) \, dx = \tan(x)\).
5. Therefore, the integral evaluates to:
\[
\tan(x) - x + C
\]
**Note:** Don't forget to add the constant of integration \(C\) at the end.
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