Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
**Problem 17: Integration of a Polynomial Exponential Product** Evaluate the integral: \[ \int x^4 e^x \, dx \] This integral involves the product of a polynomial \(x^4\) and an exponential function \(e^x\). Solving this typically requires the method of integration by parts and may involve multiple iterations due to the polynomial term. This problem is typically encountered in calculus courses, particularly those covering techniques of integration. The integral belongs to a class of integrals that can be approached using the formula for integration by parts: \[ \int u \, dv = uv - \int v \, du \] where \( u \) and \( dv \) are parts of the original integrand. For this specific integral, we would choose: \[ u = x^4 \quad \text{and} \quad dv = e^x \, dx \] Calculating each part individually: \[ du = 4x^3 \, dx \quad \text{and} \quad v = e^x \] Then, applying integration by parts iteratively will progressively decrease the power of \( x \) until the integral involves only the exponential function, which simplifies easily.
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Transcribed Image Text:**Problem 17: Integration of a Polynomial Exponential Product** Evaluate the integral: \[ \int x^4 e^x \, dx \] This integral involves the product of a polynomial \(x^4\) and an exponential function \(e^x\). Solving this typically requires the method of integration by parts and may involve multiple iterations due to the polynomial term. This problem is typically encountered in calculus courses, particularly those covering techniques of integration. The integral belongs to a class of integrals that can be approached using the formula for integration by parts: \[ \int u \, dv = uv - \int v \, du \] where \( u \) and \( dv \) are parts of the original integrand. For this specific integral, we would choose: \[ u = x^4 \quad \text{and} \quad dv = e^x \, dx \] Calculating each part individually: \[ du = 4x^3 \, dx \quad \text{and} \quad v = e^x \] Then, applying integration by parts iteratively will progressively decrease the power of \( x \) until the integral involves only the exponential function, which simplifies easily.
### Problem 16 Evaluate the integral: \[ \int e^x \arctan(e^x) \, dx \] This integral presents a function involving exponential and arctangent functions which may require advanced integration techniques such as integration by parts.
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Transcribed Image Text:### Problem 16 Evaluate the integral: \[ \int e^x \arctan(e^x) \, dx \] This integral presents a function involving exponential and arctangent functions which may require advanced integration techniques such as integration by parts.
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