Detailed handwritten notes deriving the integration by parts formula from the product rule.

I can see this is about integration by parts. The notes show the formula and start working through an example: ∫x·e^(-x) dx

Let me solve this problem step by step.

Step 1: Identify u and dv
- u = x
- dv = e^(-x) dx

Step 2: Find du and v
- du = dx (derivative of x)
- v = ∫e^(-x) dx = -e^(-x) (integral of e^(-x))

Step 3: Apply the integration by parts formula
∫u·dv = uv - ∫v·du

Substituting our values:
∫x·e^(-x) dx = x·(-e^(-x)) - ∫(-e^(-x))·dx

Step 4: Simplify
= -x·e^(-x) + ∫e^(-x) dx

Step 5: Integrate the remaining integral
∫e^(-x) dx = -e^(-x)

So we get:
= -x·e^(-x) + (-e^(-x)) + C
= -x·e^(-x) - e^(-x) + C

Step 6: Factor (optional, for cleaner answer)
= -e^(-x)(x + 1) + C

Let me verify by differentiating:
d/dx[-e^(-x)(x + 1)]
= -[d/dx(e^(-x))·(x + 1) + e^(-x)·d/dx(x + 1)]
= -[(-e^(-x))(x + 1) + e^(-x)(1)]
= -[-e^(-x)(x + 1) + e^(-x)]
= e^(-x)(x + 1) - e^(-x)
= e^(-x)·x + e^(-x) - e^(-x)
= x·e^(-x) ✓

The answer checks out!

Final Answer:
x·e^(-x) dx = -e^(-x)(x + 1) + C