Unit 8 - Objective 2 - Integral for Natural Log

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.

Problem 1:

 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].

Problem 2:
 let u = x³ + 2, then du = 3x² dx, "fix" the 3

Problem 3:
 let u = 2x + 3, then du = 2 dx, "fix" the 2

Problem 4:
 let u = tan x, then du = sec²x dx

Problem 5:

Problem 6:

 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.

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