8.2.8 Integration by Parts

Integration by Parts

Notes int_parts, AS & A Level Maths revision notes

Notes parts_alt_not, AS & A Level Maths revision notes

Notes parts_choosing, AS & A Level Maths revision notes

The hardest part is choosing u and dv/dx as there is no method for doing so
u, ideally, becomes simpler when differentiated but this is not always possible
dv/dx should be a function that can be integrated fairly easily
Be wary of functions that ‘cycle’/’repeat’ when differentiated/integrated
- e^x → e^x
- sin x → cos x → -sin x → -cos x → sin x

STEP 1: Choose u and v’, find u’ and v
STEP 2: Apply Integration by Parts
- Simplify anything straightforward
STEP 3: Do the ‘second’ integral
- If an indefinite integral remember “+c”, the constant of integration
STEP 4: Simplify and/or apply limits

Notes parts_twice, A Level & AS Level Pure Maths Revision Notes

Notes int_ln, AS & A Level Maths revision notes

A classic ‘set piece’ in almost every A level maths textbook ever written!
In general, rewriting f(x) as 1×f(x) can be a powerful problem-solving technique
This could be a question in the exam

Notes parts_def, A Level & AS Level Pure Maths Revision Notes

Always think about what an elegant, slick, professional maths solution looks like – solutions normally get more complicated at first but quickly get simpler.
If your work is continuing to get more complicated, stop and check for an error.
Try to develop a sense of ‘having gone too far down the wrong path’.
This general advice is useful to remember:
- Is the second integral harder than the first?
- Try swapping your choice of u and dv/dx
- It is rare to have to apply integration by parts more than twice

Example soltn, AS & A Level Maths revision notes