# Thread: integration by parts to evaluate the definite integral

1. ## integration by parts to evaluate the definite integral

(9t^2)(ln(t))dt at points 1 to e

2. Choose $u = \ln t$ and $dv = 9t^2 \cdot dt$.
Then $du = \frac{1}{t} \cdot dt$ and $v = 3t^3$.

Thus, calculate:

$\int u \cdot dv = uv - \int v \cdot du$

3. first thanks for your response and second I got that far but I am having trouble evaluating it at the points from 1 to e

4. Originally Posted by dorrisstroy20
first thanks for your response and second I got that far but I am having trouble evaluating it at the points from 1 to e
$\int^e _1 (9t^2\, \ln(t)) \,dt = \int^e _1 \left[3t^3\, \ln(t) - \int 3t^3 \cdot \frac{1}{t}\right] = \left[3t^3\, \ln(t) - \frac{t^2}{3}\right]^e _1 = F(e) - F(1)$

Spoiler:
$\left(3e^3\, \ln(e) - \frac{e^2}{3}\right) - \left(3(1)^3\, \ln(1) - \frac{1^2}{3}\right)$

5. So then, you calculated

$\int 9t^2 \ln t \cdot dt = 3t^3 \ln t - \int 3t^2 \cdot dt$

$\int 9t^2 \ln t \cdot dt = 3t^3 \ln t - t^3 + C$.

Hence

$\int _1 ^e 9t^2 \ln t \cdot dt = 3e^3 \ln e - e^3 - (3 \cdot 1^3 \ln 1 - 1^3)$,

by the Fundamental Theorem of Calculus. Just simplify that expression.

6. Thank you so much