Integration
| English | Chinese | Pinyin |
|---|---|---|
| integration | 积分 | jī fēn |
| fundamental theorem of calculus | 微积分基本定理 | wēi jī fēn jī běn dìng lǐ |
| constant of integration | 积分常数 | jī fēn cháng shù |
| definite integral | 定积分 | dìng jī fēn |
| integration by substitution | 换元积分法 | huàn yuán jī fēn fǎ |
Differentiation, run backwards
- If differentiating $x^3$ gives $3x^2$, then something whose derivative is $3x^2$ must be $x^3$ — or $x^3$ plus any constant.
- Integration 积分 is that reverse operation, and it is also the area under a curve.
- Those two facts being the same fact is the fundamental theorem of calculus 微积分基本定理.
The constant of integration
- Add one to the power, divide by the new power. The reverse of the power rule.
- The constant of integration 积分常数 $c$ is there because differentiating any constant gives zero, so the original constant is unrecoverable.
- $n = -1$ is excluded because it would divide by zero. That case integrates to a logarithm.
Integrate x³ with respect to x. Include the constant.
Add one to the power and divide by the new power, then add c because any constant differentiates to zero.
A definite integral needs a constant of integration.
The constant cancels in F(b) − F(a). Only an indefinite integral needs it — and there it is compulsory.
In one English sentence, explain why an indefinite integral needs "+ c".
Example: "Any constant differentiates to zero, so the original constant cannot be recovered from the derivative."
Forgetting $+c$ is the single most frequent lost mark in this module, and it is lost on questions that were otherwise answered correctly. An indefinite integral without it is not a complete answer, however good the working above it is.
The area under the curve
$\int_a^b f(x)\,dx$
A definite integral is the signed area between the curve and the axis.
Evaluate the integral of x² from 0 to 3.
[x³/3] from 0 to 3 = 27/3 − 0 = 9.
Definite integrals
- A definite integral 定积分 has limits and evaluates to a number: $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$.
- Write the square-bracket line with the limits before you substitute. It is part of the method mark.
- The constant cancels in the subtraction, which is why a definite integral needs no $+c$.
Find the area under $y = x^2$ from $x = 0$ to $x = 3$.
Three lines: the integral, the bracket with limits, the subtraction. An answer of 9 with no bracket line shows nothing about where it came from, and a marker cannot award what is not written.
Which substitution simplifies the integral of 2x(x² + 1)⁵?
Its derivative 2x is already present, so du = 2x dx and the integral becomes u⁵ du.
Substitution
- Integration by substitution 换元积分法 reverses the chain rule.
- Spot a function and its derivative both present: $\displaystyle\int 2x(x^2+1)^5\,dx$ has $x^2+1$ and its derivative $2x$.
- Substitute $u = x^2 + 1$, so $du = 2x\,dx$, and the integral becomes $\displaystyle\int u^5\,du$.
- ⚠ Change the limits too when the integral is definite, or convert back to $x$ before substituting them.
How do you check an integration answer reliably?
Differentiation undoes integration, so it is a complete check and it takes ten seconds.
Check an integral by differentiating your answer. It takes ten seconds and it is the only self-check in calculus that is completely reliable — if you do not get back what you started with, the integral is wrong.