Integration (Pure 2)
Introduced| English | Meaning |
|---|---|
| integral/ˈɪntɪɡrəl/ | A mathematical quantity obtained by accumulation, represented by a definite integral, or a family of antiderivatives represented by an indefinite integral. |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | A function in which the variable appears in an exponent, such as a fixed positive base raised to that variable. |
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | An equation involving a function and one or more of its derivatives. |
| trapezium rule/trəˈpiːzɪəm ruːl/ | A numerical method that approximates a definite integral by summing areas of trapezia beneath straight-line segments. |
| strip/strɪp/ | strip |
| estimate/ˈestɪmət/ | To find an approximate value using available information or simplifying assumptions. |
| identity/aɪˈdentɪti/ | An equation that is true for every allowed value of its variables. |
The area that eˣ can't escape
- The integral 积分 of $e^x$ is $e^x$ — the same function. This self-referential property makes exponential functions 指数函数 central to differential equations 微分方程.
- Integration techniques let you find areas, volumes, and solutions to real-world problems.
Standard integrals
Worked example. $\int e^{3x}\,dx = \dfrac{1}{3}e^{3x} + C$. The $\dfrac{1}{3}$ comes from the chain rule in reverse.
Don't forget the $\dfrac{1}{a}$. When integrating $e^{ax+b}$, you must divide by $a$. The integral of $e^{2x}$ is $\dfrac{1}{2}e^{2x}$, not $e^{2x}$.
The trapezium rule 梯形法则 adds straight-topped strips 条带 of width h to estimate 估计 the area
The area under the curve
area = ∫ f(x) dx
The integral still measures area — drag a and b to total the strip under the curve.
What is ∫ eˣ dx?
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eˣ is its own integral (and derivative): ∫eˣ dx = eˣ + C.
What is ∫ (1/x) dx?
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∫(1/x) dx = ln|x| + C (the power rule fails for n = −1).
∫ e^(2x) dx = (1/a)e^(2x) + C. What is a?
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The chain rule in reverse: divide by the coefficient of x inside the exponential.
∫ cos(3x) dx = (1/3)sin(3x) + C. Evaluate from 0 to π/6 (2 dp).
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[(1/3)sin(3x)]₀^(π/6) = (1/3)sin(π/2) − 0 = 1/3 ≈ 0.33.
Integrating powers of trig functions
- To integrate a power of $\sin$/$\cos$, use an identity 恒等式 to remove the power first.
- Example: $\sin^2 x = \dfrac{1}{2}(1 - \cos 2x)$, so $\int \sin^2 x\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$.
The trapezium rule
- When you can't integrate exactly, estimate:
- Each strip of width $h$ is a trapezium; add their areas.
The trapezium rule: divide the area into strips, approximate each as a trapezium, and sum their areas.
The trapezium rule estimates a definite integral when it cannot be found exactly.
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It approximates the area as a sum of trapezium strips of width h.
Using the trapezium rule with h = 1, estimate ∫₁³ x² dx with y-values 1, 4, 9.
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(1/2)[1 + 9 + 2(4)] = (1/2)[10 + 8] = 9. (Exact: 26/3 ≈ 8.67.)
Worked example — trapezium rule
- Estimate $\int_1^4 x^2\,dx$ using 3 strips ($h = 1$).
- $y$-values: $y_0 = 1$, $y_1 = 4$, $y_2 = 9$, $y_3 = 16$.
- $\int \approx \dfrac{1}{2}[1 + 16 + 2(4 + 9)] = \dfrac{1}{2}[17 + 26] = 21.5$.
- Exact value: $\left[\dfrac{x^3}{3}\right]_1^4 = \dfrac{64}{3} - \dfrac{1}{3} = 21$. The estimate is close.
- Integration is reverse differentiation; harder integrals use integration by substitution.
You've got it
- $\int e^{ax+b}dx = \tfrac1a e^{ax+b} + C$; $\int \tfrac{1}{ax+b}dx = \tfrac1a\ln|ax+b| + C$
- remove powers of $\sin$/$\cos$ with an identity before integrating
- the trapezium rule estimates an integral as a sum of trapezium strips