substitution
∫CALCULUS II STUDY GUIDE
Integration Techniques
Choose among substitution, integration by parts, trigonometric identities, and trigonometric substitution.
8 key ideas8 flashcards5 practice questions
Copy to my workspace →LISTEN WITH SOBATake this set with you.
Audio is saved on this device after the first play.
Read the integrand before doing any algebra.
integration by parts
identity or trig substitution
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
Substitution reverses the chain rule by replacing an inner expression and its differential.
- 02
Integration by parts reverses the product rule through ∫u dv = uv − ∫v du.
- 03
A useful choice of u becomes simpler after differentiation.
- 04
Odd powers of sine or cosine often leave one factor for substitution.
- 05
Even powers of sine and cosine often call for half-angle identities.
- 06
Products of secant and tangent depend on whether a secant pair or a secant-tangent pair can be saved.
- 07
The substitution x = a sin θ fits radicals containing a² − x².
- 08
Differentiation checks an antiderivative and catches missing factors or signs.
PART OF THE CALCULUS II GUIDE
See every set in this course and follow a focused review order.
Open the full Calculus II guide →Two ideas worth correcting now
NOT QUITE
→LIATE is a theorem that always picks u
USE THIS INSTEAD
It is a memory aid; the best choice is the one that simplifies the remaining integral.
NOT QUITE
→Every radical needs trigonometric substitution
USE THIS INSTEAD
Algebra or an ordinary substitution may be shorter.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What formula starts integration by parts?Show answer ＋
∫u dv = uv − ∫v du.
2What property makes a good choice for u?Show answer ＋
Its derivative should be simpler or fit the remaining integral.
3How can an odd power of sine be handled?Show answer ＋
Save one sine factor, convert the remaining even power with sin²θ = 1 − cos²θ, then substitute.
4How can an even power of cosine be handled?Show answer ＋
Use a half-angle identity before finding the antiderivative.
5Which substitution fits √(a²−x²)?Show answer ＋
x = a sin θ.
6Which substitution fits √(a²+x²)?Show answer ＋
x = a tan θ.
7Which substitution fits √(x²−a²)?Show answer ＋
x = a sec θ.
8How is an antiderivative checked?Show answer ＋
Differentiate it and compare the result with the original integrand.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Why should algebra come before method selection?
Factoring, division, or identity use may expose a direct substitution or a standard form.
02How do you find an antiderivative of x eˣ?
Choose u = x and dv = eˣ dx, giving x eˣ − eˣ + C.
03Why can repeated integration by parts return to the starting integral?
Some trig-exponential products cycle; after rearrangement, the repeated integral can be isolated algebraically.
04How does a trigonometric substitution remove a radical?
A Pythagorean identity converts the radical into a trigonometric expression whose sign is fixed by the chosen interval.
05What is the final step after trigonometric substitution?
Return to the original variable, often with a reference triangle or an inverse trigonometric relation.
Related study materials
〜•‿•
Ready to remember this?
Copy the set and let Soba schedule what to review next.
Study this set for free →