Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Question
![### Using a Table of Integrals with Appropriate Substitution
To solve the following integral using a table of integrals and the appropriate substitution:
\[
\int \frac{e^{3x} + 2e^{2x} + 8e^x}{\sqrt{e^{2x} - 36}} \, dx
\]
Before consulting the table of integrals, we need to make a substitution to simplify the integrand.
1. **Identify a suitable substitution:**
Let \( u = e^x \), hence \( du = e^x dx \) or \( dx = \frac{du}{u} \).
2. **Rewrite the integral in terms of \( u \):**
\[
e^{3x} = (e^x)^3 = u^3, \quad e^{2x} = (e^x)^2 = u^2, \quad e^x = u
\]
Plugging these into the integrand, we get:
\[
\int \frac{u^3 + 2u^2 + 8u}{\sqrt{u^2 - 36}} \cdot \frac{du}{u}
\]
Simplify the integrand:
\[
\int \frac{u^3 + 2u^2 + 8u}{u \sqrt{u^2 - 36}} \, du = \int \frac{u^2 + 2u + 8}{\sqrt{u^2 - 36}} \, du
\]
3. **Integrate using the appropriate entry from the table of integrals:**
The integral \(\int \frac{u^2 + 2u + 8}{\sqrt{u^2 - 36}} \, du\) can be solved by identifying an integral in the table that matches the form \(\int \frac{ax^2 + bx + c}{\sqrt{ax^2 + bx + c}} \, dx\).
4. **Look up the appropriate form in the table of integrals and solve:**
Suppose the table of integrals provides the following standard form and its solution (you may need to adjust constants):
\[
\int \frac{a u^2 + b u + c}{\sqrt{d u^2 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F541bb107-0359-491e-bc63-cc05a5ca14fa%2F56b4732f-b207-4480-84d6-2d72c9319227%2Fo403o1s_processed.png&w=3840&q=75)
Transcribed Image Text:### Using a Table of Integrals with Appropriate Substitution
To solve the following integral using a table of integrals and the appropriate substitution:
\[
\int \frac{e^{3x} + 2e^{2x} + 8e^x}{\sqrt{e^{2x} - 36}} \, dx
\]
Before consulting the table of integrals, we need to make a substitution to simplify the integrand.
1. **Identify a suitable substitution:**
Let \( u = e^x \), hence \( du = e^x dx \) or \( dx = \frac{du}{u} \).
2. **Rewrite the integral in terms of \( u \):**
\[
e^{3x} = (e^x)^3 = u^3, \quad e^{2x} = (e^x)^2 = u^2, \quad e^x = u
\]
Plugging these into the integrand, we get:
\[
\int \frac{u^3 + 2u^2 + 8u}{\sqrt{u^2 - 36}} \cdot \frac{du}{u}
\]
Simplify the integrand:
\[
\int \frac{u^3 + 2u^2 + 8u}{u \sqrt{u^2 - 36}} \, du = \int \frac{u^2 + 2u + 8}{\sqrt{u^2 - 36}} \, du
\]
3. **Integrate using the appropriate entry from the table of integrals:**
The integral \(\int \frac{u^2 + 2u + 8}{\sqrt{u^2 - 36}} \, du\) can be solved by identifying an integral in the table that matches the form \(\int \frac{ax^2 + bx + c}{\sqrt{ax^2 + bx + c}} \, dx\).
4. **Look up the appropriate form in the table of integrals and solve:**
Suppose the table of integrals provides the following standard form and its solution (you may need to adjust constants):
\[
\int \frac{a u^2 + b u + c}{\sqrt{d u^2 +
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