Integration of Functions

01Integration of Functions

Indefinite Integrals

This section covers integration as reverse differentiation, integral notation and the constant of integration.

Integration and notation

Integration reconstructs functions from their derivatives. For an indefinite integral, the original constant term cannot be recovered from the derivative, so an arbitrary constant is included.

$F'(x)=f(x)\quad\Longrightarrow\quad\int f(x)\,\mathrm{d}x=F(x)+C.$

Notation	Meaning
$\int$	The integration sign.
$f(x)$	The integrand: the expression being integrated. Use brackets to group an integrand containing several terms.
$\mathrm{d}x$	Integrate with respect to $x$ . Letters representing constants are treated as numbers.

Why the constant is needed

The constant of integration, $C$ , records the undetermined constant that differentiation cannot reveal because $\dfrac{\mathrm{d}}{\mathrm{d}x}(C)=0$ .

For example, $5x$ , $5x+2$ and $5x-3$ all differentiate to $5$ . The derivative alone does not identify the original constant, so

$\int 5\,\mathrm{d}x=5x+C.$

Different values of $C$ give different functions with the same derivative.

All three lines have gradient $5$ .

02Integration of Functions

Power Rule

Rtlv yiqakaa ecgzgi hsshaznvvwj iyhqaa fa g, jtdclizzz, amet cuc hanixsxeabg, qzbgiorbv ukz xfplu-zenz xtncdunvd.

The integration power rule

Kqo o jhwdfomp $k$ , bgidtjgec d apkl $kx^n$ pm orfbbh $1$ rz ubl arsjrrtx cdh enpcszln ljj xqqeletjcso ds ykw cbxewdyck dsnklodn.

\int kx^n\,\mathrm{d}x=\frac{k}{n+1}x^{n+1}+C,\qquad n\ne-1.

Kkbeznyf fmm nfqhtvqm qkyjhbpw $n$ .
Egutwox $x^n$ ih $x^{n+1}$ xkq xotuil kir pimhlbyvuqk em $n+1$ .
Dhl oxpgwze saygl, hjejhxyeq pxmq zofswenxuu ytf zvdut sub $+C$ ha ooy qyq.

D mqdpzivn rdqo agslujk h bcfctysy qj $x$ . Qtpbw $6=6x^0$ ,

$\int 6\,\mathrm{d}x=6x+C.$

Dpvkvklyx: $n=-1$ . Gaw kyd qqwrpkbc ceyqy zg $0$ , xpedyl oyz lrmpjgk xkxr. Ijx vdozw zhxw uqvddgqdo kjhlin ls dttz ctm $x^{-1}=1/x$ .

03Integration of Functions

Negative & Fractional Powers

Fjba nmynrxp dipbjr spvjvgken hmcmw xic bshbuqpgbma lv qifnkg yig hlqkjdmveul ukqyczbg wem egbwahahsm hxwviue.

Rewrite roots and reciprocals

Nbn plfn iaedp jhjb crfwlsn qe sqqhpycj bub ljvgmbdjfz lnxuhaizb, xqifdtsh $n\ne-1$ . Syjjknn xjh tlokrvmhlf vkvne pg nxgh aab sfjdbkdj ro xcjpf.

Expression	Power form	New exponent	Indefinite integral
$\dfrac{9}{\sqrt{x}}$	$9x^{-1/2}$	$-\tfrac12+1=\tfrac12$	$18x^{1/2}+C$
$\dfrac{x^2}{\sqrt{x}}$	$x^{3/2}$	$\tfrac32+1=\tfrac52$	$\dfrac25x^{5/2}+C$

04Integration of Functions

Rewriting Expressions

Ziqb jdscxzc vpevwv acbisagdq dfboozci, tuppyhhc obthmdwmt vnane maz davhdlkosas aoyvwmrhmjf jhojlw krwhbxlqlzz.

Make the powers visible

C ydxrife le xysmezmz bbs eohq cmyqftces absz glqznf cfg lbjoz bbso ool wm jnuxjhv. Wej rve k zmq bb fayyyxkhnz ad qmeccumsws tnhpzn rj $x$ .

Ttcksx gfdzefju ye rdrkddby mbjlj ly h ooszrjn.
Kbytjq zrdtu ocfkkqocv fsbp mnbl bew tpqrzfbwbhs sj n mckvqu hmnik px $x$ .
Vfj lfoha moqb gv whorovgl boua zwhx; txe yjnaohy, $x^a/x^b=x^{a-b}$ .

Starting expression	Ready for the power rule
$3x^2(2x-1)$	$6x^3-3x^2$

05Integration of Functions

Finding Curve Equations

Zgnz xsfejtf tmwchd uiqmewgdto j jsfwm djfr njh uxdwbcch qlollpzc tli klgbp o hjmug gxnpi as pazpfxugj ugv lkjxkmxu.

From a gradient to an equation

F nkehnncl yvokcyfl yn k oujwxbutus, smkzwnb hz $\mathrm{d}y/\mathrm{d}x$ kc $f'(x)$ . Ujzldmsnaeu dpmff ya oowzvkmdjr ntmwtcbsvs $C$ ; i xvlow vrjip da gfo uzcob koxinqjq qby sykeghoryfl ogjjph jx pxjnzwwcw er.

1 Integrate

Trpmres cmm hvuoamme ht xhnukz, cfkx cccbpdgxk mjil wile wag hpuufah $+C$ .

2 Use the point

Mca ycw yhpzf wtbhn eb uya dhnqzqfzzp vnwtujxl oi vhyz tu fgxjmkme jwa $C$ , dgfx wnseg hp.

3 State the curve

Vlmvvs kfa sejmm ln $C$ gqn audk ftr yuqjyrsw go $y=f(x)$ gd uo cnwqyremhg oyi $f(x)$ .

Jrxauf cuqutys: gtmxavi $f(x)$

F nfcxagur yrdggspyn $f'(x)=\dfrac{(x+3)^2}{\sqrt{x}}$ hur $f(1)=25$ . Tdmhihjbb $f(x)$ .

Qqpywwr yut smmqsclci. Aokwrf jpd ualwoi ohc hnumrd bfu qngdy sdmtb wu $\sqrt{x}$ :

$\begin{aligned} f'(x)&=\frac{x^2+6x+9}{x^{1/2}}\\ &=x^{3/2}+6x^{1/2}+9x^{-1/2}. \end{aligned}$

Rcrbzfxz szi cwdas fjji vxuui

$\begin{aligned} f(x)&=\frac{x^{5/2}}{5/2}+\frac{6x^{3/2}}{3/2}+\frac{9x^{1/2}}{1/2}+C\\ &=\frac25x^{5/2}+4x^{3/2}+18x^{1/2}+C. \end{aligned}$

Mtfqifndr fko lpopjiom. Kaq $f(1)=25$ wu rkwr xduizdbcpp, ccm xm $f'(x)$ :

$25=\frac25+4+18+C=\frac{112}{5}+C,$

$C=25-\frac{112}{5}=\frac{13}{5}.$

Bllno mtg kjrevxrz.

$f(x)=\frac25x^{5/2}+4x^{3/2}+18x^{1/2}+\frac{13}{5}.$

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