What is the integral from-2 to 0 of e^x(x^2+1) ?

The integral from-2 to 0 of e^x(x^2+1) is 3-(11)/(e^2)

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integral from-2 to 0 of e^x(x^2+1)

\int_{- 2}^{0} e^{x} (x^{2} + 1) d x

3 - \frac{11}{e ^{2}}

Decimal

1.51131 \dots

Solution steps

\int_{- 2}^{0} e^{x} (x^{2} + 1) d x

Apply Integration By Parts

: [(x^{2} + 1) e^{x} - \int 2 x e^{x} d x]_{- 2}^{0}

= [(x^{2} + 1) e^{x} - \int 2 x e^{x} d x]_{- 2}^{0}

\int 2 x e^{x} d x = 2 (e^{x} x - e^{x})

= [(x^{2} + 1) e^{x} - 2 (e^{x} x - e^{x})]_{- 2}^{0}

Expand

(x^{2} + 1) e^{x} - 2 (e^{x} x - e^{x}) : e^{x} x^{2} - 2 e^{x} x + 3 e^{x}

= [e^{x} x^{2} - 2 e^{x} x + 3 e^{x}]_{- 2}^{0}

Compute the boundaries:

3 - \frac{11}{e ^{2}}

= 3 - \frac{11}{e ^{2}}

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What is the integral from-2 to 0 of e^x(x^2+1) ?
The integral from-2 to 0 of e^x(x^2+1) is 3-(11)/(e^2)