Key takeaways
- An integral adds up tiny pieces to find a total, such as the area under a curve.
- Indefinite integrals give a family of functions and need + C.
- Definite integrals give a number: ∫ₐᵇ f(x) dx = F(b) − F(a).
- Power rule for integrals: ∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, for n ≠ −1.
- Check any integral by differentiating your answer.
In this guide
What is an integral?
An integral adds up a huge number of tiny pieces to find a total. On a graph, a definite integral gives the area between a curve and the x-axis.
Integration is the opposite of differentiation. A derivative breaks something into its rate of change. An integral builds the total back up from the rate.
Why do integrals matter?
Many real totals come from rates. Water flowing in litres per second, a car's speed in metres per second, electricity used in kilowatts: integrate the rate and you get the total amount, distance or energy.
Differentiation asks 'how fast?'. Integration asks 'how much altogether?'.
Definite vs indefinite integrals
| Indefinite integral | Definite integral | |
|---|---|---|
| Looks like | ∫ f(x) dx | ∫ₐᵇ f(x) dx |
| Result | A function plus a constant | A single number |
| Meaning | All antiderivatives of f | Net signed area from a to b |
| Example | ∫ x² dx = x³/3 + C | ∫₀³ x² dx = 9 |
An indefinite integral always includes + C, the constant of integration:
A definite integral uses the Fundamental Theorem of Calculus:
The basic tool is the power rule for integration: ∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, for any n except −1. Raise the power by one, then divide by the new power.
How to evaluate a definite integral step by step
- 1Find an antiderivative F(x). For powers of x, use the power rule.
- 2Substitute the upper limit b to get F(b).
- 3Substitute the lower limit a to get F(a).
- 4Subtract: F(b) − F(a). There is no + C here, because it cancels.
- 5Interpret the sign. Area below the x-axis counts as negative.
Check power integrals like this with the definite integral calculator, which also warns you when an integral does not converge.
Advanced techniques
U-substitution
U-substitution reverses the chain rule. Look for a function and its derivative both inside the integral. For ∫ 2x·(x² + 1)³ dx, let u = x² + 1, so du = 2x dx. The integral becomes ∫ u³ du = u⁴/4 + C = (x² + 1)⁴/4 + C.
Integration by parts
Integration by parts reverses the product rule. It helps when you integrate a product, like x times eˣ:
For ∫ x·eˣ dx, let u = x and dv = eˣ dx. Then du = dx and v = eˣ. So the integral is x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C. From 0 to 1, that gives (e − e) − (0 − 1) = 1.
Practise these with the u-substitution calculator, which handles integrals of the form (ax + b)ⁿ, and the integration by parts calculator, which handles xⁿ·eˣ between two limits.
Common mistakes to avoid
- Using the power rule on x⁻¹.
∫ x⁻¹ dx = ln|x| + C, notx⁰/0. - Subtracting in the wrong order. It is F(b) − F(a): upper limit first.
- Forgetting to change dx in u-substitution. Replace dx using du.
- Treating net area as total area.
∫ sin x dxfrom 0 to 2π is 0, because the part below the axis cancels the part above. For total area, split at the crossing points.
Tips and tricks
- Choose u in parts wisely. Pick u to be something that gets simpler when you differentiate it, like x or ln x.
- Trig integrals:
∫ sin(ax) dx = −cos(ax)/a + C. Try the trig integral calculator. - No antiderivative? Estimate the area with the Simpson's rule calculator.
Where you'll use it in real life
- Physics: distance from speed, and work done by a changing force.
- Engineering: areas and volumes of curved shapes.
- Economics: total profit from a marginal profit rate.
- Probability: the area under a probability curve gives a probability.
Quick summary
An integral adds up tiny pieces. Indefinite integrals give a family of antiderivatives plus C. Definite integrals give a number using ∫ₐᵇ f(x) dx = F(b) − F(a). Start with the power rule, use u-substitution to reverse the chain rule and integration by parts to reverse the product rule, and check your answer by differentiating. The free calculators linked above show every step.
Frequently asked questions
Why is there a constant of integration (+ C) in indefinite integrals?
The derivative of any constant is zero, so many functions have exactly the same derivative, each shifted up or down. The + C stands for that whole family of functions.
What does a definite integral represent geometrically?
It is the net area between the graph and the x-axis, from the lower limit to the upper limit. Area above the axis counts as positive and area below counts as negative.
When should I use u-substitution?
Use it when the integral contains a function and also the derivative of that function, such as 2x and x² + 1. Let u be the inner function.
How do I check if my integral is correct?
Differentiate your answer. If you get back the function you started with, the integral is correct.
