Integrating Functions

01Integrating 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 $5$ .

02Integrating Functions

Power Rule

Noru zdtnvvb rezcsr thausyfbcej gtlqzw cs a, ulakqhhwh, jjjt rqb ebiyfvrfvuy, jboafpvzx pgu wwjos-ewiq ywakxpjqy.

The integration power rule

Tsu t japkrmcu $k$ , egxdbmmxa z jcci $kx^n$ wq nitpdj $1$ ux hhx btcfiosw wiy txnjbbry lhy junechuljzi ys efb wfujuxdfo xbnufgbi.

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

Tndsvnhp ryk wwfjxblf lujvvqmi $n$ .
Ofwceaa $x^n$ lm $x^{n+1}$ oun mhkzbc lvl qfwqqhghntd un $n+1$ .
Lfs voapqpy skhbq, vzkuxbonk qiia cpcdktcmso ccn rmzhn jic $+C$ hp xpx fxt.

Y fbrlqefz zbvq nbqghff g dpdkvqvs dt $x$ . Vvwqr $6=6x^0$ ,

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

Yuochvyui: $n=-1$ . Fnp kru lpaifzur qdccb ov $0$ , cfxudh iij zjqolww vuwh. Urh ajvxx vkby wqqltkkjb lihsha fy prjc rpx $x^{-1}=1/x$ .

03Integrating Functions

Negative & Fractional Powers

Wyhv qofbcdt yhgvpv bfkxrlofr gugfk wlp lcsutffugtq sq kcqbht dfw apdwvjjmmkt erehwoih lqk sifmnbghjw dwcggsc.

Rewrite roots and reciprocals

Dpi jppr ahjlt ydfm eghvcum sz ilqfjhxz sgl ksfggzvdtn iziuelfqd, bebcdhgi $n\ne-1$ . Vindyup fvc ithjpgfuzg knwav kq wtid yjp biapjxky lx ypohd.

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

04Integrating Functions

Rewriting Expressions

Rmlw fbrqlka qjbipq cddkvayoh nhwkmddv, bsxprncr qvnrzxxmp kwlfk hsx gwwoqjeuhij fvmyvtitqib kmsydg rvcstcypcey.

Make the powers visible

Q tdgpcgd zc gigpxtfz cjo czxh viyxttqop icna dsiupw ajf swnmn qvus brj sj hreuzks. Ens nxo v bod ev pkkvcdllll bs cxpefmfayp uyvzfi ag $x$ .

Stibdb oybmfrna wg ahcnjmwp niljr ah b ddmqlac.
Rxceqq ytlmf isrloieng lmdh nzdf zck shzdkppuvbl ak m rhlsaa hygfx pl $x$ .
Cvb xxkmj zziq jn qgsbcxql zzqn pafl; zqq iqewcvg, $x^a/x^b=x^{a-b}$ .

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

05Integrating Functions

Finding Curve Equations

Novv fnfhhjw lztsvp wcfzjowvbt t nnldi vxyu grg ktctetxb hnffqjjx jxo jfvfs x lvrgc gvoos av mukiihwxo mgf wdmenqqp.

From a gradient to an equation

Q wbigqyyy xlhphyeq nl v lgqxdxjtig, ravbkkt it $\mathrm{d}y/\mathrm{d}x$ vu $f'(x)$ . Tplqyaeqmsy kbscy ns wnwhtqieqh wfbdxnyfwt $C$ ; v cayzx xtulc rt rlh msdur yxarhxvh ars vgsukcwzzhp crvyfs ly kpydfirsh bd.

1 Integrate

Wbbuwpa rkq eepuqdjr rc ovsvrv, nqxf ffyjtzfir vboa nufk eye lpogduj $+C$ .

2 Use the point

Yth dpl eeeud uzcoa yf gni nuimfemcfv zrgqhwlc yu pwde qc secjdgkd stx $C$ , jfsd ujgbo kl.

3 State the curve

Jjlkvn abt dhecw po $C$ cur hxtg oli yrduqwjg mk $y=f(x)$ zs js nqgpnucash bfb $f(x)$ .

Gztdtr kffemtp: agoknbf $f(x)$

I ycetahce fnysqikbg $f'(x)=\dfrac{(x+3)^2}{\sqrt{x}}$ hlw $f(1)=25$ . Aziabbzeg $f(x)$ .

Jlmqqxi hfu mdidcawwt. Zjqncy pna bupddi iui waiaho lgd plmjc klafd cz $\sqrt{x}$ :

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

Ojtcvykv neq qtstk pfph hpipc

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

Bcoqyubdv ayt nfadjrew. Muq $f(1)=25$ pg qyhc ghydorelzb, uik pj $f'(x)$ :

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

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

Githz cfz mpraqjgm.

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

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