Key takeaways
- Choose u as the inside part, then find du and replace every x and dx.
- ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ ÷ (a(n + 1)) + C, for n ≠ −1.
- The #1 mistake is forgetting to divide by a.
- Either change the limits to u-values or swap back to x before using them.
In this guide
What is u-substitution?
An integral adds up area under a curve, and you solve one by reversing a derivative. When a function has another function inside it, like (3x + 1)², the answer isn’t obvious. U-substitution gives the inside a new name, u, which turns the problem into a simple power like u².
It undoes the chain rule, which multiplies by the inside’s derivative. Substitution divides it back out.
Why does u-substitution matter?
Most integrals in science have something inside something else: (2x + 5)⁴, e^(3x), cos(5x). Substitution handles all of them with one idea, so it is usually the first method to try.
Every chain-rule derivative you have ever done can be turned into a u-substitution integral by reading it backwards.
The formula
This calculator finds a definite integral of a linear expression to a power:
| Symbol | What it means | Example |
|---|---|---|
| a | The number in front of x (cannot be 0) | 3 |
| b | The constant added inside the brackets | 1 |
| n | The power (any number except −1) | 2 |
| lo | The lower x-limit | 0 |
| hi | The upper x-limit | 1 |
How to do u-substitution step by step
- 1Choose u as the inside.
u = 3x + 1. - 2Differentiate u.
du = 3 dx, sodx = du/3. - 3Rewrite everything in u.
∫(3x + 1)² dx = (1/3)∫u² du. - 4Integrate. Using the power rule:
(1/3) × u³/3 = u³/9. - 5Swap back and use the limits.
(3x + 1)³/9. From 0 to 1:(4³ − 1³)/9 = 63/9 = 7.
Now try it yourself. Enter a, b, n and the two limits in the calculator below to see the substitution step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Mixing x-limits with u. Either change the limits to u-values or swap back to x first.
- Leaving some x behind. Every x and the dx must be replaced before you integrate.
- Using n = −1.
∫(ax + b)⁻¹ dx = ln|ax + b|/a + C, so the calculator shows an error for n = −1.
Tips and tricks
- Look for the derivative. In
∫2x(x² + 1)⁵ dx, the 2x is the derivative of x² + 1. So u = x² + 1 and the answer is(x² + 1)⁶/6 + C. - Check by differentiating.
d/dx [(3x + 1)³/9] = 3(3x + 1)² × 3/9 = (3x + 1)². Correct. - Other methods: for products like x·eˣ, use integration by parts. For plain powers of x, see the definite integral calculator.
Where you’ll use it in real life
- Physics: integrals like
∫cos(5x) dx = sin(5x)/5 + Cin waves and motion. - Probability: the normal distribution and many others need substitution.
- Engineering: stretched or shifted versions of known formulas, like (2t + 3)².
- Economics: totals from growth rates that depend on a shifted time.
Quick summary
U-substitution renames the inside part as u, replaces dx with du, and turns a hard integral into an easy one. For (ax + b)ⁿ, the result is (ax + b)ⁿ⁺¹ ÷ (a(n + 1)). Use the calculator above for any limits, with every step shown.
Worked example
A homework problem asks for ∫(3x + 1)² dx from x = 0 to x = 1 using u = 3x + 1. Here a = 3, b = 1, the power is n = 2, the lower limit is lo = 0 and the upper limit is hi = 1.
- 01∫[0 to 1] (3x + 1)^2 dx
- 02Let u = 3x + 1, then du = 3dx → dx = du/3
- 03When x=0: u=1; when x=1: u=4
- 04= (1/3) ∫ u^2 du = (1/3) · u^3/3 + C
- 05= (3x+1)^3 / (3(3)) + C
- 06F(1) = 7.1111
- 07F(0) = 0.1111
- 08Result = 7
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I choose u in u-substitution?
Pick the inside part whose derivative also appears in the integral (maybe up to a constant), such as the expression inside brackets, a square root, or an exponent.
Do I have to change the limits in u-substitution?
Only if you want to finish in terms of u. Otherwise, swap u back to x and use the original x-limits. Both give the same answer.
What is the integral of (2x + 1)³?
With u = 2x + 1 and du = 2 dx, it is (2x + 1)⁴/8 + C.
