This is the missing power, n = -1, not covered by the power rule on
worksheet 8-1. Consider the integral as a fraction where the
denominator is to the first power and the numerator is the derivative
of the denominator [du/u]. Or the numerator can be the derivative of
the denominator except for a constant factor, which can be "fixed".
The absolute value is on the natural logarithm because logs are only
defined for positive values. The domain of the log x is x > 0.
|The denominator is to the first power. Let the denominator = u = 2 + sin x. Then du = (0 + cos x) dx = cos x dx|
|This matches the numerator so it's (du/u) = ln |u| + C|
|= ln |2 + sin x| + C = ln (2 + sin x) + C|
NOTE: The answer to Example 1 above can be written without the absolute value because (2 + sin x) is always positive [sin x is always between -1 and +1, so (2 + sin x) is between +1 and +3].
|let u = x³ + 2, then du = 3x² dx, "fix" the 3|
|let u = 2x + 3, then du = 2 dx, "fix" the 2|
|let u = tan x, then du = sec²x dx|
|The denominator is NOT to the first power. This is not a log integral. The denominator is the 1/2 power so rewrite using exponents.|