integration analysis

Consider the integral

t² e^t/3 dt

This is an indefinite integral, so we need to find the most general antiderivative of the integrand f(t).

To find an antiderivative of f(t), we go through our list of integration methods:

Recognize elementary antiderivatives
Rewrite the integrand to make it easier
Use substitution to reverse the chain rule or simplify the integrand
Use integration by parts
Use inspection to see the value of a definite integral

to find one that works for this integrand. In this case, only one method is appropriate. We use integration by parts to rewrite the product of u = t² and v' = e^t/3 in the integrand using the integration by parts formula. This is shown below.

Integration by parts:
Let u = t² and v' = e^t/3. Then u' = 2t and v = 3 e^t/3, so, remembering the integration by parts formula

u v' dt = u v -

u' v dt,

we have

t² e^t/3 dt = (t²) (3 e^t/3) -

(2t) (3 e^t/3) dt = (3t² - 18t + 54) e^t/3 + C.

Note that we evaluate the integral of u' v by using another application of integration by parts:

Let u = 2t and v' = 3 e^t/3. Then u' = 2 and v = 9 e^t/3, so, remembering the integration by parts formula

u v' dt = u v -

u' v dt,

we have

(2t)(3 e^t/3) dt = (2t) (9 e^t/3) -

(2) (9 e^t/3) dt = (18t - 9) e^t/3 + C.

integration tutorial: analysis