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VED1
E L-LOVELY-H math13-1 IInd 6-8-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12

noEPk;soh dk rfDs

B'N 7H sin (log e x) + c fe;dk jZb j' ;edk j?<
∫
∫
2
2
2
2
(a) sin x cos x dx (b) cot x cosec x dx
e
(c) ∫ cos(log x ) dx (c) fJBQK ftZu'A e'Jh BjhA.
x
+
8H log 1 + e c fe;dk jZb j?<
x
+
e
(a) ∫ 1 e+ x x dx (b) ∫ 1 e x dx
x
e
−
∫
x
(c) e dx (d) ∫ 1 e+ e x x dx.
13H7 ;koKP (Summary)
• fe;h cbB dk nB[ebB gsk eoB dh fBwB fbys d' w[Zy ftXhnK jB^
(i) gqfs;EkgB d[nkok nB[ebB (Integration by Substitution)
(ii) yzv d[nkok nB[ebB (Integration by Parts)

• fJ; fefonk ftZu fdZs/ j'J/ nB[ebe (Integrand) B{z wkBe ;{soK (Standard formulae)
ftZu gfotofss eoe/ nB[eb ehsk iKdk j?.

• i/eo nB[ebe (Integrand) φ [f (x)] f ′ (x) d/ o{g dk j't/ Gkt fJj fe;h okPh f (x) d/
cbB ns/ fJ;h okPh f (x) d/ fBy/VB r[DKe f ′ (x) dk r[DBcb j't/ iK fJ; o{g ftZu
fbfynk ik ;edk j't/ sK n;hA fJ; okPh f (x) B{z t d/ pokpo wzB e/ nB[ebB eod/
jK.

13H8 Ppde'P (Keywords)

1H gqfs;EkgB (Substitution) ^ ft;EkgB.
2H ftXh (Methods) ^ sohek.

13H9 nfGnk; gqPB (Review Questions)

1H ∫ sec 3x dx dk wkB eZY'. (T[Zso L tan3x + c )
2
3
2H ∫ sin θθ dk wkB gfoefbs eo'. (T[Zso L − 3 cosθ+ 1 cos3θ+ )
3
c
d
4 12
3
2
3H cbB cos x sin x dk x d/ ;kg/y nB[ebB gsk eo'. (T[Zso L − cos x + c )
3

4H ∫ sin 2x+ π   dx dk wkB gsk eo' (T[Zso L sin 2x + c )

 2  2
−
1
-1
2
5H ∫ 4sin x dx dk wkB eZY' (T[Zso L 2 (sin x) + c)
−
(1 x 2 )

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