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2.5 Integration

Syllabus
9709–2028–2029
Topic
2.5
Level
AS

Learning objectives

2.5.1Extended integration rules• extend the idea of 'reverse differentiation' to include the integration of eax + b, ax b 1 +, sin(ax + b), cos(ax + b) and sec2(ax + b) Knowledge of the general method of integration by substitution is not required.2.5.2Trig integration• use trigonometrical relationships in carrying out integration e.g. use of double-angle formulae to integrate sin2 x or cos2(2x).2.5.3Trapezium rule• understand and use the trapezium rule to estimate the value of a definite integral. Including the use of sketching graphs in simple cases to determine whether the trapezium rule gives an over-estimate or an under-estimate.

Reverse five linear-inner derivatives with the $1/a$ factor

| Integrand ($a

e0$) Antiderivative
eax+be^{ax+b} eax+b/a+Ce^{ax+b}/a+C
1/(ax+b)1/(ax+b) (1/a)ln⁡∣ax+b∣+C(1/a)\ln|ax+b|+C
sin⁡(ax+b)\sin(ax+b) −cos⁡(ax+b)/a+C-\cos(ax+b)/a+C
cos⁡(ax+b)\cos(ax+b) sin⁡(ax+b)/a+C\sin(ax+b)/a+C
sec⁡2(ax+b)\sec^2(ax+b) an(ax+b)/a+Can(ax+b)/a+C

Match the whole integrand to one row, keep the linear inner expression unchanged, divide by its gradient aa, carry any outside constant, and include +C+C for an indefinite integral.

\int 3e^{2x-1},dx= rac32e^{2x-1}+C,\int rac{5}{3x+4},dx= rac53\ln|3x+4|+C.

Differentiate the answer: the chain factor aa must cancel the inserted 1/a1/a. For definite integrals, use an interval that does not cross a point where the integrand is undefined.

General integration by substitution and integration by parts are not required in Pure Mathematics 2; use these direct reverse-derivative patterns only.

Reduce squared trig functions before integrating

\sin^2u= rac{1-\cos2u}{2},\qquad \cos^2u= rac{1+\cos2u}{2}.These follow from the two useful forms of $\cos2u$.

First rewrite the squared sine or cosine as a constant plus/minus a double-angle cosine. Then integrate term by term using the linear-inner rule, including the factor created by the doubled angle.

\int\sin^2x,dx=\int rac{1-\cos2x}{2},dx= rac x2- rac{\sin2x}{4}+C.

\int\cos^2(2x),dx=\int rac{1+\cos4x}{2},dx= rac x2+ rac{\sin4x}{8}+C.

The square is on the trig value, so the ordinary power integration rule does not apply. This objective uses trig identities with the direct P2 antiderivatives, not a general substitution method.

Weight trapezium ordinates and use chord position to judge error

For $n$ equal strips, $h=(b-a)/n$ and ordinates $y_0,\ldots,y_n$:T= rac h2\left[y_0+y_n+2(y_1+\cdots+y_{n-1})
ight].

Confirm equal spacing, list all n+1n+1 ordinates in order, weight endpoints once and internal ordinates twice, then retain appropriate accuracy because this is an estimate.

Sketch on each strip Chord relative to curve Estimate
concave up chord above curve over-estimate
concave down chord below curve under-estimate

With $h=0.5$ at $x=0,0.5,1$, there are two strips and three ordinates:T=0.25(y_0+2y_1+y_2).

The number of ordinates is one more than the number of strips. If curvature changes, inspect or split the graph rather than claiming one global error direction from a single segment.