| Law | Expression |
|---|---|
| Product | \(\log_{a}(xy)=\log_{a}x+\log_{a}y\) |
| Quotient | \(\log_{a}\!\left(\dfrac{x}{y}\right)=\log_{a}x-\log_{a}y\) |
| Power | \(\log_{a}x^{k}=k\log_{a}x\) |
| Change of base | \(\log_{a}x=\dfrac{\log_{b}x}{\log_{b}a}\) |
| Exponential | \(a^{\log_{a}x}=x\) |
Typical exam problem – solve \(2^{x}=5\):
\(\displaystyle x=\frac{\log 5}{\log 2}\) (common log) or \(x=\frac{\ln5}{\ln2}\).
Example – differentiate \(y=\dfrac{x^{2}\,e^{3x}}{1+\sin x}\).
Apply product & quotient rules → final answer
\(y'=\dfrac{(2x\,e^{3x}+3x^{2}e^{3x})(1+\sin x)-x^{2}e^{3x}\cos x}{(1+\sin x)^{2}}\).
For data points \((x_{0},y_{0}),\dots,(x_{n},y_{n})\) equally spaced by \(h\):
\[ \int_{x_{0}}^{x_{n}}f(x)\,dx\;\approx\;\frac{h}{2}\Bigl(y_{0}+2\sum_{i=1}^{n-1}y_{i}+y_{n}\Bigr). \]Typical exam task – estimate \(\displaystyle\int_{0}^{2}x^{3}\,dx\) using \(h=1\). Result: \(\frac{1}{2}\bigl(0+2(1)+8\bigr)=5\) (exact value \(=4\)).
Use when the integrand contains a function \(g(x)\) and its derivative \(g'(x)\).
Example \(\displaystyle\int e^{3x+2}\,dx\):
\(u=3x+2,\;du=3dx\Rightarrow dx=\frac{du}{3}\) → \(\frac13\int e^{u}du=\frac13e^{u}+C=\frac13e^{3x+2}+C\).
Based on the product rule:
\[ \int u\,dv = uv-\int v\,du. \]Choose \(u\) (to differentiate) and \(dv\) (to integrate) using the LIATE hierarchy (Log → Inv‑trig → Algebraic → Trig → Exp).
Example \(\displaystyle\int x\,e^{2x}\,dx\):
\(u=x,\;du=dx,\;dv=e^{2x}dx,\;v=\tfrac12e^{2x}\) → \(\tfrac12xe^{2x}-\tfrac14e^{2x}+C\).
Applicable to proper rational functions \(\displaystyle\int\frac{P(x)}{Q(x)}dx\) where \(\deg P<\deg Q\).
| Factor of \(Q(x)\) | Decomposition |
|---|---|
| Simple linear \((x-a)\) | \(\displaystyle\frac{A}{x-a}\) |
| Repeated linear \((x-a)^{k}\) | \(\displaystyle\frac{A_{1}}{x-a}+\frac{A_{2}}{(x-a)^{2}}+\dots+\frac{A_{k}}{(x-a)^{k}}\) |
| Irreducible quadratic \((x^{2}+bx+c)\) | \(\displaystyle\frac{Bx+C}{x^{2}+bx+c}\) |
| Repeated quadratic \((x^{2}+bx+c)^{k}\) | \(\displaystyle\frac{B_{1}x+C_{1}}{x^{2}+bx+c}+\dots+\frac{B_{k}x+C_{k}}{(x^{2}+bx+c)^{k}}\) |
Worked example (repeated linear & quadratic):
\[ \int\frac{3x^{2}+5x+2}{(x-1)^{2}(x^{2}+2x+5)}dx. \]Decompose:
\[ \frac{3x^{2}+5x+2}{(x-1)^{2}(x^{2}+2x+5)}= \frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{Cx+D}{x^{2}+2x+5}. \]Multiply through, equate coefficients (or substitute convenient \(x\) values) to obtain \(A=1,\;B=2,\;C=-1,\;D=0\). Hence
\[ \int\Bigl(\frac{1}{x-1}+\frac{2}{(x-1)^{2}}-\frac{x}{x^{2}+2x+5}\Bigr)dx = \ln|x-1|- \frac{2}{x-1}-\tfrac12\ln(x^{2}+2x+5)+C. \]Integrals of \(\displaystyle\int\sin^{m}x\cos^{n}x\,dx\).
Example \(\displaystyle\int\sin^{3}x\cos^{2}x\,dx\) (odd \(\sin\)):
\[ \sin^{3}x=(1-\cos^{2}x)\sin x\;\Rightarrow\; \int(1-\cos^{2}x)\cos^{2}x\sin x\,dx \overset{u=\cos x}{=} -\int(u^{2}-u^{4})du = \frac{\cos^{5}x}{5}-\frac{\cos^{3}x}{3}+C. \]Used for radicals of the form \(\sqrt{a^{2}\pm x^{2}}\) or \(\sqrt{x^{2}-a^{2}}\).
| Radical | Substitution | Resulting identity |
|---|---|---|
| \(\sqrt{a^{2}-x^{2}}\) | \(x=a\sin\theta\) | \(\sqrt{a^{2}-x^{2}}=a\cos\theta\) |
| \(\sqrt{a^{2}+x^{2}}\) | \(x=a\tan\theta\) | \(\sqrt{a^{2}+x^{2}}=a\sec\theta\) |
| \(\sqrt{x^{2}-a^{2}}\) | \(x=a\sec\theta\) | \(\sqrt{x^{2}-a^{2}}=a\tan\theta\) |
Example \(\displaystyle\int\frac{dx}{\sqrt{9-x^{2}}}\):
\[ x=3\sin\theta,\;dx=3\cos\theta d\theta,\;\sqrt{9-x^{2}}=3\cos\theta \;\Longrightarrow\; \int d\theta=\theta+C =\arcsin\!\left(\frac{x}{3}\right)+C. \]| Technique | Typical syllabus form | Key step |
|---|---|---|
| Substitution | \(\displaystyle\int f(g(x))g'(x)\,dx\) | Set \(u=g(x)\) |
| Integration by Parts | \(\displaystyle\int u\,dv\) | Choose \(u, dv\) via LIATE |
| Partial Fractions | \(\displaystyle\int\frac{P(x)}{Q(x)}dx\) (proper) | Decompose \(Q(x)\) into linear/quadratic factors |
| Trigonometric Integrals | \(\displaystyle\int\sin^{m}x\cos^{n}x\,dx\) | Use parity, double‑angle, and substitution |
| Trig Substitution | \(\displaystyle\int\frac{dx}{\sqrt{a^{2}\pm x^{2}}}\) etc. | Replace \(x\) by \(a\sin\theta, a\tan\theta\) or \(a\sec\theta\) |
Region bounded by \(y=f(x)\) and the x‑axis on \([a,b]\), rotated about the x‑axis:
\[ V=\pi\int_{a}^{b}[f(x)]^{2}\,dx. \]If rotating about the y‑axis, express the curve as \(x=g(y)\) and integrate w.r.t. \(y\):
\[ V=\pi\int_{c}^{d}[g(y)]^{2}\,dy. \]Outer curve \(y=f(x)\), inner curve \(y=g(x)\) (\(f\ge g\)), rotation about the x‑axis:
\[ V=\pi\int_{a}^{b}\bigl([f(x)]^{2}-[g(x)]^{2}\bigr)\,dx. \]For rotation about the y‑axis, use \(x=h(y)\) (outer) and \(x=k(y)\) (inner):
\[ V=\pi\int_{c}^{d}\bigl([h(y)]^{2}-[k(y)]^{2}\bigr)\,dy. \]Best when the axis of rotation is parallel to the integration direction.
Region bounded by \(y=x^{2}\) and \(y=2x\) rotated about the y‑axis.
Washer method (express \(x\) as functions of \(y\)):
Shell method (vertical strips): radius \(x\), height \(2x-x^{2}\), \(x\in[0,2]\).
\[ V=2\pi\int_{0}^{2}x\,(2x-x^{2})dx =2\pi\int_{0}^{2}(2x^{2}-x^{3})dx =2\pi\Bigl[\frac{2x^{3}}{3}-\frac{x^{4}}{4}\Bigr]_{0}^{2} =\frac{8\pi}{3}. \]| Method | When most convenient | Typical formula |
|---|---|---|
| Disk | Region expressed as \(x=g(y)\) with no hole. | \(V=\pi\displaystyle\int_{c}^{d}[g(y)]^{2}dy\) |
| Washer | Region between two curves \(x=h(y)\) (outer) and \(x=k(y)\) (inner). | \(V=\pi\displaystyle\int_{c}^{d}\bigl([h(y)]^{2}-[k(y)]^{2}\bigr)dy\) |
| Shell | Vertical strips are simpler; axis of rotation is vertical. | \(V=2\pi\displaystyle\int_{a}^{b}x\,\bigl[\text{height}(x)\bigr]dx\) |
A first‑order ODE is separable if it can be written as
\[ \frac{dy}{dx}=g(x)\,h(y). \]Example \(\displaystyle\frac{dy}{dx}=x\,y^{2}\), with \(y(0)=1\).
\[ \frac{1}{y^{2}}\,dy = x\,dx \;\Longrightarrow\; -\frac{1}{y}= \frac{x^{2}}{2}+C. \] Using \(y(0)=1\) gives \(C=-1\). Hence \[ -\frac{1}{y}= \frac{x^{2}}{2}-1 \;\Longrightarrow\; y=\frac{1}{1-\frac{x^{2}}{2}}. \]Useful results:
Line through \(P(1,2,3)\) with direction \(\mathbf{d}= \langle 2,-1,2\rangle\). Find distance from \(Q(4,0,5)\) to the line.
\[ \mathbf{PQ}= \langle 3,-2,2\rangle,\qquad \text{distance}= \frac{|\mathbf{PQ}\times\mathbf{d}|}{|\mathbf{d}|} = \frac{\sqrt{( -2\cdot2-2\cdot(-1))^{2}+(2\cdot2-3\cdot2)^{2}+(3\cdot(-1)-(-2)\cdot2)^{2}}}{\sqrt{2^{2}+(-1)^{2}+2^{2}}} = \frac{\sqrt{36}}{3}=2. \]| Identity | Form |
|---|---|
| Reciprocal | \(\csc\theta=1/\sin\theta,\;\sec\theta=1/\cos\theta,\;\cot\theta=1/\tan\theta\) |
| Pythagorean | \(\sin^{2}\theta+\cos^{2}\theta=1\), \(1+\tan^{2}\theta=\sec^{2}\theta\), \(1+\cot^{2}\theta=\csc^{2}\theta\) |
| Double‑angle | \(\sin2\theta=2\sin\theta\cos\theta\), \(\cos2\theta=\cos^{2}\theta-\sin^{2}\theta\) |
| Half‑angle | \(\sin^{2}\theta=\tfrac12(1-\cos2\theta)\), \(\cos^{2}\theta=\tfrac12(1+\cos2\theta)\) |
| R‑formula | \(a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)\), \(R=\sqrt{a^{2}+b^{2}}\) |
| Law | Expression |
|---|---|
| Product | \(\log_{a}(xy)=\log_{a}x+\log_{a}y\) |
| Quotient | \(\log_{a}\!\left(\dfrac{x}{y}\right)=\log_{a}x-\log_{a}y\) |
| Power | \(\log_{a}x^{k}=k\log_{a}x\) |
| Change of base | \(\log_{a}x=\dfrac{\log_{b}x}{\log_{b}a}\) |
| Exponential inverse | \(a^{\log_{a}x}=x\) |
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