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Chapter 4: Problem 36

Determine these indefinite integrals. $$\int \frac{4}{5} e^{-10 x} d x$$

Short Answer

Expert verified
-\frac{2}{25} e^{-10x} + C

Step by step solution

01

- Identify the integral form

Recognize that the integral \(\frac{4}{5} e^{-10 x} dx\) fits the standard form \(a \cdot e^{b x} dx\), where \(a = \frac{4}{5}\) and \(b = -10\).
02

- Set up the integral

Set up the integral using the known formula for exponentials. The integral of \(e^{bx} dx\) is \(\frac{1}{b} e^{bx} + C\).
03

- Apply the formula

Since \(a = \frac{4}{5}\) and \(b = -10\), apply the formula: \[\frac{4}{5} \int e^{-10x} dx = \frac{4}{5} \left( \frac{1}{-10} e^{-10x} \right) + C\].
04

- Simplify the expression

Simplify the expression inside the integral: \[\frac{4}{5} \left( \frac{1}{-10} e^{-10x} \right) + C = -\frac{4}{50} e^{-10x} + C = -\frac{2}{25} e^{-10x} + C\].

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Key Concepts

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

exponential function
An exponential function is a mathematical function in the form of \[e^{bx}\].
An exponential function has the constant 'e' (approximately equal to 2.71828), raised to a variable exponent.
The general format is \[f(x) = a \times e^{bx}\], where:
  • a is a constant multiplier.
  • b is another constant which affects the growth rate.
Exponential functions grow very quickly.
In our exercise, the term \[e^{-10x}\] represents the exponential part, with 10 being the rate at which the function grows (or decays, since it鈥檚 negative).
Understanding the structure helps us to integrate it properly.
integration
Integration is the process of finding the integral of a function.
It鈥檚 essentially the reverse operation of differentiation.
When we integrate a function, we are looking for a function whose derivative matches the original function.
Indefinite integrals don鈥檛 have upper and lower limits. They involve a constant term 'C', representing an unknown constant of integration.
In our exercise, we used the formula for the integral of an exponential function:\[ \frac{1}{b} e^{bx} + C\].
Here:
  • We identified \(e^{-10x}\) as the exponential part of our function, with \(b = -10\).
  • We determined the overall integral by multiplying the constant coefficient \( \frac{4}{5} \) and then integrating the exponential term.
Understanding this formula allows us to handle various exponential integrals efficiently.
calculus step-by-step
Calculus involves working through problems in a series of logical steps.
This step-by-step approach ensures you accurately apply concepts and reach correct solutions.
Let's break down our solved exercise:
  • Step 1 - Identify the integral form:
    We recognized the integral \(\frac{4}{5} e^{-10x} dx\) matches the standard format \(a \times e^{bx} dx\).
  • Step 2 - Set up the integral:
    We used the known integral formula for an exponential function: \( \frac{1}{b} e^{bx} + C\).
  • Step 3 - Apply the formula:
    We placed our values of \(a\) and \(b\), specifically \(a = \frac{4}{5}\) and \(b = -10\).
  • Step 4 - Simplify the expression:
    We simplified it as \( -\frac{2}{25} e^{-10x} + C\).
Following these clear steps ensures accuracy and understanding in solving similar integral problems.

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Most popular questions from this chapter

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