Substitution reverses the Chain Rule. It works when the integrand contains a function and its derivative together.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Choosing u
Take u to be the inside of a composition, usually what sits under a power or inside a function.
Handling du
Differentiate to get du = u′ dx. The remaining part of the integrand must match, up to a constant.
Constants are adjustable
A missing constant factor can be inserted and compensated for. A missing variable factor cannot.
Definite integrals
Either change the limits to u values, or convert back to x before evaluating. Mixing the two is the classic error.
Check by differentiating
Every integral is checkable. Differentiate the answer and it should return the integrand.
The chain rule, run backwards
Substitution recognises an integrand as the derivative of a composition and undoes it. Spotting that the integrand contains an inner function and its derivative is the whole of the technique.
Change the limits or change back
With a definite integral, either convert the limits to u-values or convert back to x before evaluating. Substituting the original limits into the u-expression is a frequent and serious error.
Choosing u
Pick the inner function of a composition, or whatever sits inside a bracket, root or exponent. If its derivative appears elsewhere in the integrand up to a constant, the substitution will work.
Check by differentiating
Differentiate the answer and confirm it returns the integrand. Since integration is defined as the reverse of differentiation, this check is complete and takes a few seconds.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Second point(3, 9)
- Slope between them3
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 Marcus Substitute
Marcus integrates 2x(x² + 1)³ with respect to x.
- Step 1
The inside of the composition is x² + 1, so he takes u = x² + 1.
Step 4: Your Turn
Practice makes it stick.
The Choice
Problem 1 of 2
For the integral of 2x(x² + 1)³, what should u be? Enter 1 for x² + 1, 2 for 2x.
The Derivative
Problem 2 of 2
If u = x² + 1, then du = kx dx. What is k?
Substitute and Integrate
1 of 8
If u = x³ + 2, then du = kx² dx. What is k?
2 of 8
If u = 5x + 1, then du = k dx. What is k?
3 of 8
Integral of u³ du gives u⁴ ÷ k. What is k?
4 of 8
Integral of u⁵ du gives u⁶ ÷ k. What is k?
5 of 8
Can a missing constant factor be compensated for? 1 yes, 0 no.
6 of 8
Can a missing variable factor be compensated for? 1 yes, 0 no.
7 of 8
Put the substitution process in order.
- 1Differentiate to find du.
- 2Rewrite the whole integral in terms of u.
- 3Integrate, then convert back or change the limits.
- 4Choose u as the inside of the composition.
8 of 8
If u = x² − 7, then du = kx dx. What is k?
Step 5: Quick Check
Show what you know.
Question 1 of 2
If u = x⁴ + 3, then du = kx³ dx. What is k?
Question 2 of 2
What must be done with the limits in a definite integral?
What You Learned
- Substitution reverses the Chain Rule.
- Take u as the inside of the composition and match du to the rest.
- For a definite integral, change the limits or convert back before evaluating.