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
9
![### Evaluate the following indefinite integral
\[
\int \left( \frac{1 + 8t^7}{5t} \right) dt
\]
\[
\int \left( \frac{1 + 8t^7}{5t} \right) dt = \boxed{\phantom{__________}} \text{(Type an exact answer.)}
\]
### Explanation
The task involves evaluating the given indefinite integral. The expression inside the integral is \(\frac{1 + 8t^7}{5t}\), which can be simplified further before integrating.
To begin, let's split the fraction:
\[
\frac{1 + 8t^7}{5t} = \frac{1}{5t} + \frac{8t^7}{5t}
\]
This simplifies to:
\[
\frac{1}{5t} + \frac{8t^6}{5}
\]
Now, integrate each term separately:
\[
\int \left( \frac{1}{5t} + \frac{8t^6}{5} \right) dt
\]
### Integration Steps:
1. **Integrate \( \frac{1}{5t} \)**
\[
\int \frac{1}{5t} dt = \frac{1}{5} \int \frac{1}{t} dt = \frac{1}{5} \ln|t|
\]
2. **Integrate \( \frac{8t^6}{5} \)**
\[
\int \frac{8t^6}{5} dt = \frac{8}{5} \int t^6 dt = \frac{8}{5} \left( \frac{t^7}{7} \right) = \frac{8t^7}{35}
\]
### Combining the results:
\[
\int \left( \frac{1}{5t} + \frac{8t^6}{5} \right) dt = \frac{1}{5} \ln|t| + \frac{8t^7}{35} + C
\]
### Final Answer:
\[
\boxed{\frac{1}{5} \ln|t| + \frac{8t^7}{35} + C}
\]
Here, \(C\) is the constant of integration](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02b4a1f4-9e35-42db-aa2a-674bb26df94b%2F52c36215-fdd0-44be-ab75-2c598bdb88d9%2F5llcm8b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluate the following indefinite integral
\[
\int \left( \frac{1 + 8t^7}{5t} \right) dt
\]
\[
\int \left( \frac{1 + 8t^7}{5t} \right) dt = \boxed{\phantom{__________}} \text{(Type an exact answer.)}
\]
### Explanation
The task involves evaluating the given indefinite integral. The expression inside the integral is \(\frac{1 + 8t^7}{5t}\), which can be simplified further before integrating.
To begin, let's split the fraction:
\[
\frac{1 + 8t^7}{5t} = \frac{1}{5t} + \frac{8t^7}{5t}
\]
This simplifies to:
\[
\frac{1}{5t} + \frac{8t^6}{5}
\]
Now, integrate each term separately:
\[
\int \left( \frac{1}{5t} + \frac{8t^6}{5} \right) dt
\]
### Integration Steps:
1. **Integrate \( \frac{1}{5t} \)**
\[
\int \frac{1}{5t} dt = \frac{1}{5} \int \frac{1}{t} dt = \frac{1}{5} \ln|t|
\]
2. **Integrate \( \frac{8t^6}{5} \)**
\[
\int \frac{8t^6}{5} dt = \frac{8}{5} \int t^6 dt = \frac{8}{5} \left( \frac{t^7}{7} \right) = \frac{8t^7}{35}
\]
### Combining the results:
\[
\int \left( \frac{1}{5t} + \frac{8t^6}{5} \right) dt = \frac{1}{5} \ln|t| + \frac{8t^7}{35} + C
\]
### Final Answer:
\[
\boxed{\frac{1}{5} \ln|t| + \frac{8t^7}{35} + C}
\]
Here, \(C\) is the constant of integration
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