Math Problem Statement

Evaluate the integral \( \int x^2 e^x \, dx \)

Solution

To evaluate the integral

x2exdx\int x^2 e^x \, dx

we can use the technique of integration by parts multiple times. Let’s proceed with the steps:

Step 1: Integration by Parts (First Application)

For integration by parts, we use the formula:

udv=uvvdu\int u \, dv = uv - \int v \, du

Choose:

  • u=x2u = x^2 (which means du=2xdxdu = 2x \, dx)
  • dv=exdxdv = e^x \, dx (which gives v=exv = e^x)

Then, x2exdx=x2ex2xexdx\int x^2 e^x \, dx = x^2 e^x - \int 2x e^x \, dx

Step 2: Integration by Parts (Second Application)

Now, we need to evaluate the integral 2xexdx\int 2x e^x \, dx.

Choose:

  • u=2xu = 2x (which means du=2dxdu = 2 \, dx)
  • dv=exdxdv = e^x \, dx (which gives v=exv = e^x)

Then, 2xexdx=2xex2exdx\int 2x e^x \, dx = 2x e^x - \int 2 e^x \, dx

Step 3: Final Integration

Now we integrate 2exdx\int 2 e^x \, dx, which is straightforward:

2exdx=2ex\int 2 e^x \, dx = 2 e^x

Putting it All Together

Substituting back, we get:

x2exdx=x2ex(2xex2ex)\int x^2 e^x \, dx = x^2 e^x - (2x e^x - 2 e^x)

Simplifying, we find:

x2exdx=x2ex2xex+2ex+C\int x^2 e^x \, dx = x^2 e^x - 2x e^x + 2 e^x + C

where CC is the constant of integration.


Would you like more details, or do you have any questions?

Related Questions

  1. How does integration by parts work, and when is it useful?
  2. What are alternative methods for evaluating integrals like this one?
  3. How would the answer change if the integral was xnexdx\int x^n e^x \, dx for any integer nn?
  4. Could substitution help simplify similar integrals?
  5. How can we confirm the result by differentiating the answer?

Tip

When using integration by parts repeatedly, keep track of terms carefully to avoid errors in substitution.

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Math Problem Analysis

Mathematical Concepts

Integration
Integration by Parts
Exponential Functions

Formulas

Integration by parts formula: \( \int u \, dv = uv - \int v \, du \)

Theorems

Integration by Parts

Suitable Grade Level

Undergraduate Calculus