What is the integral from 0 to ln(3) of x^2e^{3x} ?

The integral from 0 to ln(3) of x^2e^{3x} is 9ln^2(3)-6ln(3)+52/27

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integral from 0 to ln(3) of x^2e^{3x}

\int_{0}^{l n (3)} x^{2} e^{3 x} d x

9 ln^{2} (3) - 6 ln (3) + \frac{52}{27}

Decimal Notation

6.19679 \dots

Solution steps

\int_{0}^{l n (3)} x^{2} e^{3 x} d x

Apply Integration By Parts

: [\frac{1}{3} (e^{3 x} x^{2} - 3 \cdot \int \frac{2}{3} e^{3 x} x d x)]_{0}^{l n (3)}

= [\frac{1}{3} (e^{3 x} x^{2} - 3 \cdot \int \frac{2}{3} e^{3 x} x d x)]_{0}^{l n (3)}

\int \frac{2}{3} e^{3 x} x d x = \frac{2}{27} (3 e^{3 x} x - e^{3 x})

= [\frac{1}{3} (e^{3 x} x^{2} - 3 \cdot \frac{2}{27} (3 e^{3 x} x - e^{3 x}))]_{0}^{l n (3)}

3 \cdot \frac{2}{27} (3 e^{3 x} x - e^{3 x}) = \frac{2}{9} (3 e^{3 x} x - e^{3 x})

= [\frac{1}{3} (e^{3 x} x^{2} - \frac{2}{9} (3 e^{3 x} x - e^{3 x}))]_{0}^{l n (3)}

Compute the boundaries:

9 ln^{2} (3) - 6 ln (3) + \frac{52}{27}

= 9 ln^{2} (3) - 6 ln (3) + \frac{52}{27}

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What is the integral from 0 to ln(3) of x^2e^{3x} ?
The integral from 0 to ln(3) of x^2e^{3x} is 9ln^2(3)-6ln(3)+52/27