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Sheet 3 · IntegrationAI-marked, provisional

2(b)

Find $\displaystyle\int x e^{2x}\,dx$ .

3/5

Let $u = x$ and $dv = e^{2x}\,dx$ , so

$v = 2e^{2x}$

$\int x e^{2x}\,dx = 2xe^{2x} + \int 2e^{2x}\,dx$

$= 2xe^{2x} + e^{2x} + C$

1
Careless slip · Algebra or sign slip
$v = 2e^{2x}$
Differentiate your $v$ . Do you get back $e^{2x}$ ?
2
Structural mistake · Method
$\int x e^{2x}\,dx = 2xe^{2x} + \int 2e^{2x}\,dx$
Write out the integration by parts formula first. Which sign sits in front of $\int v\,du$ ?

What you do each week

Upload: Drop in the sheet as a PDF. It is split into questions, each tagged with its topics.
Attempt: Type your working or photograph it. If you are stuck, take a nudge, then an approach, then a skeleton.
Get marked: A score out of 5, every mistake boxed, and a hint for each one rather than the answer.
Revise: Before exams, retry what went wrong and try fresh variants of your weakest topics.
Spot the error: Hunt for the slips in a worked solution written with your own common errors planted in it.