QuestionEvaluate the definite integral. $$\int_{1}^{2}\left(e^{3 u}-\frac{1}{(u+2)^{2}}\right) d u$$

Studdy Solution
Simplify the expression:
$$\frac{1}{3} e^{6} + \frac{1}{4} - \frac{1}{3} e^{3} - \frac{1}{3}$$
Combine like terms:
$$\frac{1}{3} (e^{6} - e^{3}) + \left(\frac{1}{4} - \frac{1}{3}\right)$$
Calculate the constant term:
$$\frac{1}{4} - \frac{1}{3} = -\frac{1}{12}$$
Final expression:
$$\frac{1}{3} (e^{6} - e^{3}) - \frac{1}{12}$$
The value of the definite integral is:
$$\boxed{\frac{1}{3} (e^{6} - e^{3}) - \frac{1}{12}}$$