Compute [math: ∫01(2x2+3x−2)ex dx]\displaystyle \int _0^1{(2x^2+3x-2)e^x dx}.

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BlackTom AI · Accepted Answer

We start by considering the definite integral: ∫ from 0 to 1 of (2x^2 + 3x − 2) e^x dx.

Step 1: Apply integration by parts to separate the polynomial from e^x. Let u = 2x^2 + 3x − 2 and dv = e^x dx. Then du = (4x + 3) dx and v = e^x.

Thus ∫_0^1 (2x^2 + 3x − 2) e^x dx = [ (2x^2 + 3x − 2) e^x ]_0^1 − ∫_0^1 (4x + 3) e^x dx.

Step 2: Evaluate the boundary term: (2x^2 + 3x − 2) e^x at x = 1 and x = 0.
- At x = 1: (2(1)^2 + 3(1) − 2) e^1 = (2 + 3 − 2) e = 3e.
- At x = 0: (2(0)^2 + 3(0) − 2) e^0 = (−2) · 1 = −2.
So the boundary term is 3e − (−2) = 3e + 2.

Step 3: Compute the remaining integral ∫_0^1 (4x + 3) e^x dx using integration by parts again. Let u = 4x + 3 and dv = e^x dx, so du = 4 dx and v = e^x.
Then ∫ (4x + 3) e^x dx = (4x + 3) e^x − ∫ 4 e^x dx = (4x + 3) e^x − 4 e^x = (4x − 1) e^x.
Evaluate from 0 to 1: [(4x − 1) e^x]_0^1 = (4·1 − 1) e^1 − (4·0 − 1) e^0 = 3e − (−1) = 3e + 1.

Step 4: Put everything together.
∫_0^1 (2x^2 + 3x − 2) e^x dx = (boundary term) − (second integral) = (3e + 2) − (3e + 1) = 1.

Thus the computed value of the integral is 1.

_MATH1013_1ABCD_2025 Subsection 5.2 (closed on 15 Nov)

Compute [math: ∫01(2x2+3x−2)ex dx]\displaystyle \int _0^1{(2x^2+3x-2)e^x dx}.

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