→ Use the Table of Integrals to find the following antiderivatives. Be sure to cite the Formula number of the integral in the Tables (eg. #81). dx x²-4 √1 + x²dx=
→ Use the Table of Integrals to find the following antiderivatives. Be sure to cite the Formula number of the integral in the Tables (eg. #81). dx x²-4 √1 + x²dx=
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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![# Table of Integrals
### Trigonometric Forms
1. **Integral of \(\sin^n u\)**:
\[
\int \sin^n u \, du = -\frac{1}{n} \sin^{n-1} u \cos u + \frac{n - 1}{n} \int \sin^{n-2} u \, du
\]
2. **Integral of \(\cos^n u\)**:
\[
\int \cos^n u \, du = \frac{1}{n} \cos^{n-1} u \sin u + \frac{n - 1}{n} \int \cos^{n-2} u \, du
\]
3. **Sum and Difference of Sine and Cosine Integrals**:
- \(\int \sin au \sin bu \, du\)
- \(\int \cos au \cos bu \, du\)
- \(\int \sin au \cos bu \, du\)
These involve sine and cosine functions of different arguments and are integrated using the identities:
\[
\int \sin au \sin bu \, du = \frac{\sin (a - b) u}{2(a - b)} - \frac{\sin (a + b) u}{2(a + b)} + C
\]
\[
\int \cos au \cos bu \, du = \frac{\sin (a - b) u}{2(a - b)} + \frac{\sin (a + b) u}{2(a + b)} + C
\]
\[
\int \sin au \cos bu \, du = -\frac{\cos (a - b) u}{2(a - b)} + \frac{\cos (a + b) u}{2(a + b)} + C
\]
4. **Inverse Trigonometric Integrals**:
- Integrals like \(\int \sin^{-1} u \, du\) and \(\int \cos^{-1} u \, du\) are given, with results involving inverse trigonometric functions and square roots.
### Key Integrals and Their Results
- **\(\int \tan^n u \, du\) and \(\int \cot^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8df10466-bb36-424d-be07-c1f915d82bc7%2Fc0f82d0e-c39e-463a-88bb-9d84d627672b%2Fvmbv573_processed.png&w=3840&q=75)
Transcribed Image Text:# Table of Integrals
### Trigonometric Forms
1. **Integral of \(\sin^n u\)**:
\[
\int \sin^n u \, du = -\frac{1}{n} \sin^{n-1} u \cos u + \frac{n - 1}{n} \int \sin^{n-2} u \, du
\]
2. **Integral of \(\cos^n u\)**:
\[
\int \cos^n u \, du = \frac{1}{n} \cos^{n-1} u \sin u + \frac{n - 1}{n} \int \cos^{n-2} u \, du
\]
3. **Sum and Difference of Sine and Cosine Integrals**:
- \(\int \sin au \sin bu \, du\)
- \(\int \cos au \cos bu \, du\)
- \(\int \sin au \cos bu \, du\)
These involve sine and cosine functions of different arguments and are integrated using the identities:
\[
\int \sin au \sin bu \, du = \frac{\sin (a - b) u}{2(a - b)} - \frac{\sin (a + b) u}{2(a + b)} + C
\]
\[
\int \cos au \cos bu \, du = \frac{\sin (a - b) u}{2(a - b)} + \frac{\sin (a + b) u}{2(a + b)} + C
\]
\[
\int \sin au \cos bu \, du = -\frac{\cos (a - b) u}{2(a - b)} + \frac{\cos (a + b) u}{2(a + b)} + C
\]
4. **Inverse Trigonometric Integrals**:
- Integrals like \(\int \sin^{-1} u \, du\) and \(\int \cos^{-1} u \, du\) are given, with results involving inverse trigonometric functions and square roots.
### Key Integrals and Their Results
- **\(\int \tan^n u \, du\) and \(\int \cot^
Transcribed Image Text:### Example Problem: Finding Antiderivatives
Use the Table of Integrals to find the following antiderivatives. Be sure to cite the formula number of the integral in the Tables (e.g., #81).
1. \(\int \frac{dx}{x^2 - 4} =\)
2. \(\int \sqrt{1 + x^2} \, dx =\)
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