How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. I = ∫ e³ˣ sin x dx. To find: I.
2 · Write the rule and choose the first function. ∫ u v dx = u ∫ v dx − ∫ [u′ ∫ v dx] dx. Take u = sin x and v = e³ˣ (the ILATE memory aid puts T before E).
3 · Apply the rule once. I = sin x × e³ˣ/3 − ∫ cos x × e³ˣ/3 dx = (1/3) e³ˣ sin x − (1/3) ∫ e³ˣ cos x dx.
4 · Apply the rule again to the new integral. ∫ e³ˣ cos x dx = cos x × e³ˣ/3 − ∫ (−sin x) e³ˣ/3 dx = (1/3) e³ˣ cos x + (1/3) I.
5 · Substitute back. I = (1/3) e³ˣ sin x − (1/9) e³ˣ cos x − (1/9) I.
6 · Solve for I. (10/9) I = (1/9) e³ˣ (3 sin x − cos x), so I = (1/10) e³ˣ (3 sin x − cos x).
7 · Check and box the answer. I = (e³ˣ/10)(3 sin x − cos x) + C.
Scoring tip: Keep the same type of first function (the trig one) in both rounds. If you switch, the integral just cancels and you learn nothing.
