Math Problem Statement

Find the antiderivative

Find the antiderivative of f(x) = (1 + xe^x) / x

Solution

To find the antiderivative of the function

f(x)=1+xexx,f(x) = \frac{1 + xe^x}{x},

we'll start by simplifying the function. We can rewrite it by separating terms:

f(x)=1x+ex.f(x) = \frac{1}{x} + e^x.

Now we can integrate each term individually.

  1. Integrate 1x\frac{1}{x}: 1xdx=lnx.\int \frac{1}{x} \, dx = \ln |x|.

  2. Integrate exe^x: exdx=ex.\int e^x \, dx = e^x.

Combining these results, the antiderivative of f(x)f(x) is:

f(x)dx=lnx+ex+C,\int f(x) \, dx = \ln |x| + e^x + C,

where CC is the constant of integration.

Would you like a more detailed explanation, or do you have any specific questions?


Here are five related questions that expand on this topic:

  1. How do you find the antiderivative of a function that is not easily separable?
  2. What are the techniques for integrating functions involving exponential terms?
  3. How does the constant of integration affect the family of functions for an antiderivative?
  4. What is the significance of the natural logarithm function in integration?
  5. How can we verify if our antiderivative is correct?

Tip: When integrating, always consider simplifying the function first to make the integration process easier.

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Math Problem Analysis

Mathematical Concepts

Calculus
Antiderivative
Integration

Formulas

∫(1/x) dx = ln|x|
∫e^x dx = e^x

Theorems

Integration by separation of terms

Suitable Grade Level

Grades 11-12