3 √(²x³ + ²³²-e²2²x + 4) dx X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![The image displays a definite integral expression as follows:
\[
\int \left( 2x^5 + \frac{3}{x} - e^{-2x} + 4 \right) \, dx
\]
This expression represents the indefinite integral of the function \(2x^5 + \frac{3}{x} - e^{-2x} + 4\) with respect to \(x\). This integral involves basic polynomial terms, a rational function, an exponential function, and a constant, each of which needs to be integrated separately. Here’s a brief look at each component:
1. **Polynomial Term:** \(2x^5\) integrates to \(\frac{2}{6}x^6 = \frac{1}{3}x^6\).
2. **Rational Function:** \(\frac{3}{x}\) integrates to \(3 \ln |x|\).
3. **Exponential Function:** \(-e^{-2x}\) integrates to \(\frac{1}{2}e^{-2x}\), following the rule for integrating exponential functions.
4. **Constant Term:** The constant \(4\) integrates to \(4x\).
These individual integrations can be summed up to find the antiderivative of the original function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F939f19e0-6626-477a-ba7a-4282434e880a%2Fdf9ae334-a7ff-4d78-b8fe-c83b01d8f5f2%2F9i2w2cm_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays a definite integral expression as follows:
\[
\int \left( 2x^5 + \frac{3}{x} - e^{-2x} + 4 \right) \, dx
\]
This expression represents the indefinite integral of the function \(2x^5 + \frac{3}{x} - e^{-2x} + 4\) with respect to \(x\). This integral involves basic polynomial terms, a rational function, an exponential function, and a constant, each of which needs to be integrated separately. Here’s a brief look at each component:
1. **Polynomial Term:** \(2x^5\) integrates to \(\frac{2}{6}x^6 = \frac{1}{3}x^6\).
2. **Rational Function:** \(\frac{3}{x}\) integrates to \(3 \ln |x|\).
3. **Exponential Function:** \(-e^{-2x}\) integrates to \(\frac{1}{2}e^{-2x}\), following the rule for integrating exponential functions.
4. **Constant Term:** The constant \(4\) integrates to \(4x\).
These individual integrations can be summed up to find the antiderivative of the original function.
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