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Magazine Chapter 7: Problem 36
Use integration by parts to evaluate each integral. \(\int z a^{z} d z\)
Short Answer
Expert verified
\(\int z a^z \, dz = \frac{z a^z}{\ln(a)} - \frac{a^z}{(\ln(a))^2} + C.\)
Step by step solution
01
Identify Parts
In the integration by parts formula \( \int u \, dv = uv - \int v \, du \), choose \( u = z \) and \( dv = a^z \, dz \). This identifies the functions to differentiate and integrate respectively.
02
Differentiate and Integrate Parts
Differentiate \( u = z \) to get \( du = dz \). Integrate \( dv = a^z \, dz \) to find \( v = \frac{a^z}{\ln(a)} \) (assuming \( a > 0 \) and \( a eq 1 \)).
03
Apply Integration by Parts Formula
Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula: \( \int z a^z \, dz = \frac{z a^z}{\ln(a)} - \int \frac{a^z}{\ln(a)} \, dz \).
04
Solve Remaining Integral
The remaining integral is \( \int \frac{a^z}{\ln(a)} \, dz \), which simplifies to \( \frac{1}{\ln(a)} \int a^z \, dz = \frac{1}{\ln(a)} \cdot \frac{a^z}{\ln(a)} + C \), because \( \int a^z \, dz = \frac{a^z}{\ln(a)} \).
05
Combine and Simplify
Combine the results to get: \[\int z a^z \, dz = \frac{z a^z}{\ln(a)} - \frac{a^z}{(\ln(a))^2} + C.\] This is the solution to the integral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
Definite integrals play a crucial role in calculus, providing a method to calculate the area under a curve between two limits. A definite integral differs from an indefinite integral by having specific boundaries, usually denoted as the lower and upper limits. The notation for a definite integral over a function \( f(x) \) from \( a \) to \( b \) is \( \int_{a}^{b} f(x) \, dx \). This integral will give a numerical value that represents the accumulated area under the curve \( f(x) \) from \( x = a \) to \( x = b \).
- Integration by Parts:** This technique helps evaluate integrals by breaking them into simpler parts. It's particularly useful when dealing with a product of functions.
- Application:** For example, finding the area under a curve such as \( z a^z \) would involve setting specific boundaries and then evaluating the definite integral using methods like integration by parts.
Differentiation
Differentiation is a fundamental concept in calculus, focusing on calculating the rate at which a function changes. For a given function \( f(x) \), its derivative \( f'(x) \) provides information on how \( f(x) \) increases or decreases as \( x \) changes.
- Slope of a Curve:** Differentiation helps us determine the slope, or steepness, of a function at any given point.
- Product Function:** When differentiating a product as in \( z a^z \), we often use techniques like the product rule or, in integration by parts, assign a part to differentiate.
Exponential Functions
Exponential functions are characterized by a constant base raised to a variable exponent, typically in the form \( a^x \). They are essential in describing growth and decay processes in natural and social sciences.
- Unique Growth:** Exponential functions grow or decay at rates proportional to their value, leading to rapid increases or decreases.
- Integration and Differentiation:** Calculating integrals and derivatives of exponential functions often involves the natural logarithm \( \ln(a) \) if the base \( a \) is different from \( e \), the natural number for which \( \ln(e) = 1 \).
