Check the techniques in order of cost. A basic antiderivative is cheapest, then substitution, then parts or partial fractions.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Substitution
Look for a function and its derivative both present. That pairing is the signal.
Parts
A product of unrelated kinds of function, such as a polynomial times an exponential, calls for integration by parts.
Partial fractions
A rational function with a factorable denominator calls for decomposition.
Try algebra first
Expanding, splitting a fraction, or a trigonometric identity often turns a hard integral into a basic one.
Always verifiable
Whichever route you take, differentiating the answer confirms it. No integration answer needs to be uncertain.
A checklist for integrals
Try substitution first, then parts, then partial fractions, then trigonometric identities. Recognising which the integral wants is the actual skill, and having an order stops you guessing.
What the integrand signals
An inner function with its derivative present suggests substitution. A product of unlike functions suggests parts. A rational function with a factorable denominator suggests partial fractions.
Rewriting often does the work
Algebraic simplification, splitting a fraction, or a trigonometric identity can turn an intimidating integral into an elementary one. Reaching for a technique before simplifying is a common waste of effort.
Differentiate to check
The answer can always be verified by differentiating it. Given how many steps an integration can take, this check is worth the seconds it costs on every problem.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Second point(2, 4)
- Slope between them2
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Ines Pick a Technique
Ines faces the integral of x times e^(x²).
- Step 1
It is a product, which might suggest integration by parts.
Step 4: Your Turn
Practice makes it stick.
The Signal
Problem 1 of 2
A function and its derivative are both present. Which technique? 1 substitution, 2 parts.
The Rational
Problem 2 of 2
A rational function with a factorable denominator. Which technique? 1 partial fractions, 2 parts.
Which Technique
1 of 8
x times eˣ. Which technique? 1 substitution, 2 parts.
2 of 8
x times e^(x²). Which technique? 1 substitution, 2 parts.
3 of 8
1 over ((x−1)(x−2)). Which technique? 1 partial fractions, 2 parts.
4 of 8
ln x on its own. Which technique? 1 substitution, 2 parts.
5 of 8
Can every integration answer be checked by differentiating? 1 yes, 0 no.
6 of 8
Which is cheapest to try first? 1 substitution, 2 partial fractions.
7 of 8
Sort each integral by the technique it wants.
Tap something to move it.
- Empty
- Empty
8 of 8
2x times (x² + 1)³. Which technique? 1 substitution, 2 parts.
Step 5: Quick Check
Show what you know.
Question 1 of 2
x times sin x. Which technique? 1 substitution, 2 parts.
Question 2 of 2
What signals substitution?
What You Learned
- Try the cheapest technique first: basic form, then substitution, then parts or partial fractions.
- Substitution needs a function and its derivative both present.
- Every answer is checkable by differentiating.