Unit 7: Strategy for Integration
No new formula here — just the pattern-reading that decides which technique an integral is actually asking for.
4 topics · 16 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Reading an Integral to Find Its Technique
- How it worksHow do the shape of the integrand, the presence of a product, and any square roots each point to a technique?
- Worked exampleFind the integral of x cos(x) dx by recognizing it as a job for integration by parts.
- Worked exampleFind the integral of x/√(x2 + 9) dx by recognizing it as a job for plain substitution.
- Common mistakeWhy doesn't seeing a square root in an integral automatically mean trig substitution is required?
Your turn1 of 2
Which technique is the right first move for the integral of x/√(x2 + 9) dx?Integrals That Need Two Techniques in a Row
- How it worksHow does simplifying with an initial substitution reveal a second, different technique hiding underneath?
- Worked exampleFind the integral of e√x dx using a substitution followed by integration by parts.
- Common mistakeWhy doesn't picking the first technique that seems to apply guarantee an integral is finished?
- Going furtherWhat happens when an integral needs a trig substitution followed by a trig integral technique on top?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 3 e√x, satisfying F(0) = 0.When Substitution Alone Isn't Enough
- How it worksHow does a mismatched power between the polynomial and the square root signal that trig substitution is needed?
- Worked exampleFind the integral of x2 √(x2 - 4) dx using trig substitution.
- Common mistakeWhy doesn't letting u equal the expression under a square root always simplify an integral?
- Ties togetherHow does spotting a mismatched power connect back to the general strategy of reading an integral first?
Your turn1 of 2
Find the definite integral of x2 √(x2 - 4) dx from 4 to 6, using trig substitution.Integrals with No Elementary Antiderivative
- How it worksHow do mathematicians know for certain some antiderivatives cannot be written in elementary form at all?
- Worked exampleShow me why the integral from 0 to 1 of e−x^2 dx has to be evaluated numerically rather than symbolically.
- Common mistakeWhy doesn't the absence of an elementary antiderivative mean a definite integral has no value at all?
- Going furtherWhat options are left for evaluating an integral once every symbolic technique has failed?
Your turn1 of 2
Which of these integrals has NO elementary antiderivative (cannot be written as a finite combination of polynomials, exponentials, logs, and trig functions)?