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Chapter 6: Problem 30

Use integration by parts to find each integral. $$ \int(x+b) e^{a x} d x \quad(a \neq 0) $$

Short Answer

Expert verified
\( \frac{1}{a}(x+b) e^{ax} - \frac{1}{a^2} e^{ax} + C \)

Step by step solution

01

Choose Functions for Integration by Parts

We start by identifying the two parts of the integral: let \( u = x + b \) and \( dv = e^{ax} \, dx \). This choice allows us to simplify the integral when applying the formula for integration by parts.
02

Differentiate and Integrate Functions

Differentiate \( u \) to find \( du \) and integrate \( dv \) to find \( v \):\[ du = dx \] \[ v = \frac{1}{a} e^{ax} \]
03

Apply the Integration by Parts Formula

The integration by parts formula is \( \int u \, dv = uv - \int v \, du \). Substitute the values we have:\[ \int (x+b) e^{ax} \, dx = \left( x+b \right) \left( \frac{1}{a} e^{ax} \right) - \int \left( \frac{1}{a} e^{ax} \right) \, dx \]
04

Calculate the Remaining Integral

The remaining integral is \( \int \frac{1}{a} e^{ax} \, dx \), which simplifies to:\[ \frac{1}{a^2} e^{ax} + C \] (Where \( C \) is the constant of integration).
05

Write the Final Expression

Combine the results to provide the solution:\[ \int (x+b) e^{ax} \, dx = \frac{1}{a}(x+b) e^{ax} - \frac{1}{a^2} e^{ax} + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
Definite integrals are a core concept in calculus used to calculate the exact area under a curve between two points. Unlike indefinite integrals, which give a family of functions plus a constant, definite integrals compute a specific numerical value. This value represents the total accumulation, like area or mass, over an interval. To evaluate a definite integral such as \( \int_a^b f(x) \, dx \):
  • Begin by finding the indefinite integral, which is the antiderivative of \( f(x) \).
  • Use the fundamental theorem of calculus, which states that if \( F(x) \) is the antiderivative of \( f(x) \), then \( \int_a^b f(x) \, dx = F(b) - F(a) \).
  • This gives the net area, accounting for sections above and below the x-axis.
Definite integrals are crucial for solving problems involving physical quantities and probabilities, as these often require precise measurements over specific intervals.
Exponential Functions
Exponential functions, characterized by the form \( f(x) = e^{ax} \), are powerful mathematical expressions where the variable is the exponent. The base \( e \) is an irrational constant approximately equal to 2.71828, known for its unique properties in calculus. Key properties of exponential functions include:
  • Rapid growth or decay, depending on the sign of the exponent \( a \).
  • The derivative of \( e^{ax} \) is straightforward: \( \frac{d}{dx} e^{ax} = a e^{ax} \).
  • The integral is similarly direct: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \).
  • Used in various fields, from population modeling in biology to calculating compound interest in finance.
Their continuous nature and well-defined rates of change make exponential functions central to differential equations and natural growth models.
Differentiation
Differentiation is a fundamental process in calculus used to find the rate at which a function changes at any point, also known as the derivative. This process allows us to understand how functions behave and is essential in optimization problems and motion analysis. To differentiate a function, follow these steps:
  • Apply the power rule for polynomial terms: \( \frac{d}{dx} x^n = n x^{n-1} \).
  • Utilize the product rule when differentiating the product of two functions: if \( u(x) \) and \( v(x) \) are functions, then \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).
  • Use the chain rule for composite functions: if a function \( y = g(f(x)) \), then \( \frac{dy}{dx} = g'(f(x)) f'(x) \).
Differentiation is utilized across sciences and engineering to model behaviors and predict future outcomes by understanding current trends and changes.

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Most popular questions from this chapter

Evaluate each improper integral or state that it is divergent. \(\int_{1}^{\infty} \frac{1}{x^{0.99}} d x\)

Manufacturers estimate the upper limit for sales of digital cameras to be 25 million annually and find that sales increase in proportion to both current sales and the difference between the sales and the upper limit. In 2005 sales were 6 million, and in 2008 were 20 million. Find a formula for the annual sales (in millions) \(t\) years after 2005 . Use your answer to predict sales in \(2012 .\)

Evaluate each improper integral or state that it is divergent. \(\int_{0}^{x} \frac{x^{2}}{x^{3}+1} d x\)

Show that the size of the permanent endowment needed to generate an annual \(C\) dollars forever at interest rate \(r\) compounded continuously is \(\mathrm{C} / \mathrm{r}\) dollars.

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