The antiderivative of xe^x is a fundamental concept in calculus, closely related to other mathematical functions such as xe^x, e^x, and the logarithmic function ln(x). Understanding its properties and applications is crucial for solving complex integration problems, exploring differential equations, and analyzing various mathematical models. This article delves into the antiderivative of xe^x, elucidating its derivation, properties, and significance in various scientific disciplines.
The Antiderivative of xe^x
To find the antiderivative of xe^x, you’ll need to use the technique of integration by parts. Here’s a step-by-step guide:
Step 1: Choose u and dv
- Let
u = x
and
dv = e^x dx
Step 2: Find du and v
- “`
du = dx
and
v = e^x.
**Step 3: Apply integration by parts**
* The formula for integration by parts is:
∫u dv = uv – ∫v du
Plugging in our values of u, dv, du, and v, we get:
∫xe^x dx = (x)(e^x) – ∫e^x dx
**Step 4: Solve the remaining integral**
* The remaining integral, ∫e^x dx, is a straightforward integral. Its antiderivative is:
∫e^x dx = e^x + C.
**Step 5: Substitute the results back into the main equation**
* Plugging the antiderivative of the remaining integral back into the main equation, we get:
∫xe^x dx = (x)(e^x) – (e^x + C) = xe^x – e^x + C.
“`
Additional Notes:
- The constant C is a constant of integration, which can take any real value.
- The antiderivative of xe^x can also be found using other integration techniques, such as substitution or partial fractions.
Table Summarizing the Steps:
| Step | Action |
|---|---|
| 1 | Choose u and dv |
| 2 | Find du and v |
| 3 | Apply integration by parts |
| 4 | Solve the remaining integral |
| 5 | Substitute the results back into the main equation |
Question 1:
- What is the concept of the antiderivative of xe^x?
Answer:
- The antiderivative of xe^x is a function whose derivative is xe^x.
Question 2:
- How can we apply integration by parts to find the antiderivative of xe^x?
Answer:
- To find the antiderivative of xe^x using integration by parts, we let u = x and dv = e^x dx.
Question 3:
- What is the significance of the constant of integration in the antiderivative of xe^x?
Answer:
- The constant of integration in the antiderivative of xe^x represents the arbitrary value that is added to the solution to account for the indefinite integral.
Well, there you have it, folks! The antiderivative of xe^x has been cracked. I know it might not be the most exciting thing in the world, but hey, it’s math, and math is cool. If you’re a bit of a math wizard and want to dive deeper, feel free to check out other mind-boggling integrations. And don’t forget to come back for more math adventures! Thanks for swinging by, and catch you next time!