Reverse the product rule.
Walk through
Step 1 of 5
Recognise a product
∫ x · eˣ dx is a product of two unlike things — no substitution will fix it. This calls for integration by parts: ∫ u dv = uv − ∫ v du.
Integration by parts is the product rule run backwards — it trades one integral for another, hoping the new one is easier.
When parts shines
- ∫ x sin x dx, ∫ x² eˣ dx — a polynomial times a sine/cosine/exponential (apply parts repeatedly).
- ∫ ln x dx — sneaky: take u = ln x, dv = dx, giving x ln x − x + C.
- ∫ eˣ sin x dx — apply parts twice and solve for the original integral algebraically.
Your turn
∫ x cos x dx = ?
By parts
Try it
∫x · eˣ dx
u = x, dv = eˣ dx. uv − ∫v du = x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C.