|The power rule for integrals
can be used for more than
just algebraic functions.
Finding the derivative of trigonometric, exponential, and logarithmic functions leads to more choices for u and du.
|Remember:||1) sin³ is the same as (sin x)³, (ln³)x is the same as (ln x)³.|
|2) To use the integral formula above, when you identify the u,
the integral must also contain its derivative, du.
tan x sec²x dx
Method I: Since the derivative of the tangent is secant², let u = tan x. Then let du = sec²x dx.
Method II: Let u = sec x, then du = sec x tan x dx. To include "du" we would have to think of the integral written as:
Both of these answers are correct, even though they do not look the same. By taking the derivative of each answer, you can see that they both have the same derivative. By using trig identities, you can show that these two antiderivatives only differ by a constant.