L&T, 16. (some overlap).
We shall consider only one type
The coefficient of must be negative, if it is positive we need a different approach which involves hyperbolic functions (not discussed here). The method is as follows
Example 5.25:
Calculate .
Solution:
Complete the square: . Equate the coefficients of each power. : , contains no unknowns. : (therefore ). The constant term gives , and thus , , and
We substitute , , which leads to
(The integral is a standard integral and can be found in the tables, but is easily checked by using the chain rule and
)