INTEGRALI INDEFINITI / ESERCIZI PROPOSTI
INTEGRALI INDEFINITI / ESERCIZI PROPOSTI
INTEGRALI INDEFINITI / ESERCIZI PROPOSTI
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ANALISI MATEMATICA I - A.A. 2011/2012<br />
<strong>INTEGRALI</strong> <strong>INDEFINITI</strong> / <strong>ESERCIZI</strong> <strong>PROPOSTI</strong><br />
L’asterisco contrassegna gli esercizi più difficili.<br />
1. Calcolare i seguenti integrali usando la linearità dell’integrale:<br />
<br />
x 2 − 3 x 2 +3 <br />
a)<br />
x 5 dx..................................................... log |x| + 9<br />
4x<br />
+ c <br />
4<br />
x 2 − 4<br />
b)<br />
x − 2 dx. ................................................................. 1<br />
2 x2 +2x + c <br />
3x 3 − 3<br />
c)<br />
x − 1 dx. ........................................................... x 3 + 3 2 x2 +3x + c <br />
<br />
x<br />
d) dx. ..............................................................[x − log |x +1| + c]<br />
1+x<br />
x 3 + x 2 +1<br />
e)<br />
dx. ......................................... 1<br />
x − 1<br />
3 x3 + x 2 +2x +3log|x − 1| + c <br />
<br />
cos 2 x<br />
f)<br />
dx. ................................................................[x +cosx + c]<br />
1+sinx<br />
<br />
g) tan 2 xdx. ..................................................................[tan x − x + c]<br />
2. Calcolare i seguenti integrali immediati usando la regola di integrazione per sostituzione:<br />
<br />
a) e sin x cos xdx. .................................................................. e sin x + c <br />
<br />
<br />
b) cos 5 x sin xdx. ............................................................... − cos6 x<br />
6<br />
+ c<br />
cos (log x)<br />
c)<br />
dx. ..............................................................[sin (log x)+c]<br />
x<br />
<br />
1<br />
d)<br />
x √ log x dx. ................................................................. 2 √ log x + c <br />
<br />
1<br />
e) √ dx................................................................ √ <br />
2<br />
3x +1<br />
3 3x +1+c<br />
<br />
1<br />
f)<br />
cos 2 (5x +9) dx. ...................................................... 1<br />
5<br />
tan (5x +9)+c<br />
<br />
g) x 3 sin x 4 <br />
cos x4<br />
dx. ............................................................... −<br />
4<br />
+ c<br />
<br />
<br />
<br />
h) xe 2x2−1 1<br />
dx. .................................................................<br />
−1 4 e2x2 + c<br />
<br />
i) x 1+x 2 dx...................................................... √ 1<br />
3 1+x<br />
2<br />
1+x 2 + c <br />
<br />
x 2<br />
l) √ dx. ............................................................. √<br />
− 2<br />
1 − x<br />
3 3 1 − x3 + c <br />
<br />
x<br />
m)<br />
1+x 4 dx. ............................................................... 1<br />
2 arctan x2 + c <br />
1
2 M.GUIDA, S.ROLANDO<br />
3. Calcolare i seguenti integrali usando la regola di integrazione per parti:<br />
<br />
a) x 2 log xdx. .......................................................... <br />
1<br />
3 x3 log x − 1 3 + c<br />
<br />
b) x 2 cos xdx. .................................................. x 2 − 2 sin x +2x cos x + c <br />
<br />
x<br />
c) x 2 e 2x <br />
dx. ........................................................... 2 − x + 1 e<br />
2x<br />
2 2 + c<br />
<br />
d) x cosh (3x) dx. ................................................ 1<br />
3 x sinh 3x − 1 9<br />
cosh 3x + c<br />
<br />
<br />
<br />
x<br />
e) (x +5)logxdx. ...........................................<br />
2<br />
2 +5x log x − x2<br />
4 − 5x + c<br />
log x<br />
<br />
<br />
f)<br />
x 2 dx. ................................................................... − 1+log x<br />
x<br />
+ c<br />
<br />
g) arctan xdx. ............................................... x arctan x − 1 2 log 1+x 2 + c <br />
<br />
h) arcsin xdx. ..................................................... x arcsin x + √ 1 − x 2 + c <br />
<br />
<br />
<br />
i) x cos 2 1 x sin 2x cos 2x<br />
xdx. .................................................<br />
2 2<br />
+<br />
4<br />
+ x2<br />
2<br />
+ c<br />
<br />
l) e 2x cos xdx. ........................................................... sin x+2 cos x<br />
5<br />
e 2x + c <br />
4. Calcolare i seguenti integrali di funzioni razionali:<br />
<br />
x +2<br />
<br />
<br />
a)<br />
x 2 − 4x +4 dx. .................................................... log |x − 2| − 4<br />
x−2 + c<br />
<br />
2x +5<br />
b)<br />
x 2 +2x − 3 dx. ........................................... 7<br />
4 log |x − 1| + 1 4<br />
log |x +3| + c<br />
<br />
x +1<br />
<br />
<br />
c)<br />
x 2 − 2x +4 dx. .................................. 1<br />
2 log(x2 − 2x +4)+ √ 2<br />
3<br />
arctan x−1 √<br />
3<br />
+ c<br />
<br />
x +1<br />
<br />
<br />
d)<br />
x 2 − x +5 dx. ................................... 1<br />
2 log(x2 − x +5)+ √ 3<br />
19<br />
arctan 2x−1 √<br />
19<br />
+ c<br />
<br />
2x 2 + x<br />
<br />
e)<br />
(x +2)(x 2 +2x +6) dx. ............ log |x +2| + 1 2 log x 2 +2x +6 <br />
− √ 4<br />
5<br />
arctan x+1 √<br />
5<br />
+ c<br />
<br />
3x − 1<br />
<br />
<br />
f)<br />
(x − 1) (x − 2) 2 dx. ................................................ 2log x−1<br />
x−2<br />
− 5<br />
x−2 + c<br />
x 4 − 5x 3 +8x 2 − 9x +11<br />
<br />
<br />
x<br />
g)<br />
x 2 dx. .....................<br />
3<br />
− 5x +6<br />
3 +2x − log |x − 2| +2log|x − 3| + c<br />
<br />
1<br />
<br />
<br />
h)<br />
(x 2 +1) 2 dx. .................................................... 1 x<br />
2 1+x<br />
+ 1 2 2 arctan x + c<br />
<br />
1<br />
<br />
<br />
i)<br />
(x 2 +1) 3 dx. .................................................. 1 3x 3 +5x<br />
8<br />
+ 3 (x 2 +1) 2 8 arctan x + c<br />
<br />
x 2<br />
<br />
<br />
l)<br />
(x +1) 4 dx. ............................................................ − 1 3x 2 +3x+1<br />
3<br />
+ c<br />
(x+1) 3<br />
5. Calcolare i seguenti integrali mediante opportune sostituzioni:<br />
e x +1<br />
a)<br />
e 2x +1 dx............................................. x − 1 2 log e 2x +1 +arctane x + c <br />
<br />
(5e x +4)e x<br />
b)<br />
(e x − 2) (e 2x + e x +1) dx. .......................... 2log(e x − 2) − log e 2x + e x +1 + c
c)<br />
d)<br />
e)<br />
f)<br />
g)<br />
h)<br />
<br />
<br />
<br />
<br />
<br />
<br />
x √ 1 − xdx. ....................................<br />
<strong>INTEGRALI</strong> <strong>INDEFINITI</strong> 3<br />
<br />
2<br />
5 (1 − √ x)2 1 − x − 2 3 (1 − x) √ <br />
1 − x + c<br />
1+ √ x<br />
x (1 + 3√ x) dx. ...............................[6 6√ x +logx − log (1 + 3√ x) − arctan 6√ x + c]<br />
1<br />
cos x dx. ................................................................ log 1+sin x cos x<br />
+ c <br />
1<br />
<br />
2sinx +cosx − 1 dx................................................ 1 <br />
2 log tan(x/2)<br />
tan(x/2)−2<br />
+ c<br />
cos 2 x<br />
<br />
1 − 2sin 2 x dx. ................................................ 1 <br />
4 log sin x+cos x<br />
sin x−cos x + 1 2 x + c<br />
tan 3 xdx. ............................................... 1<br />
2 tan2 x − 1 2 log 1+tan 2 x + c <br />
6. Calcolare i seguenti integrali mediante la sostituzione suggerita a fianco:<br />
x2<br />
a) − 1 dx, x =cosht...................................... √<br />
x<br />
2 x2 − 1 − 1 2 cosh−1 x + c <br />
x2 <br />
x<br />
b) +1dx, x =sinht....................................<br />
2 (1 + x2 )+ 1 2 sinh−1 x + c<br />
<br />
1 − x<br />
c*)<br />
1+x dx, x =sint. ........................................... arcsin x + √ 1 − x 2 + c <br />
<br />
1<br />
<br />
d)<br />
(x 2 +1) √ x 2 +1 dx, x =sinht. ............................................. √ 1<br />
+ c x 2 +1<br />
√<br />
x2 +1<br />
<br />
e)<br />
x 2 dx, x =sinht. .......................................... sinh −1 x − √ <br />
x 2 +1<br />
x<br />
+ c<br />
√<br />
x2 − 1<br />
f)<br />
dx, t = x<br />
x<br />
2 − 1. ............................. √ x 2 − 1 − arctan √ x 2 − 1 + c <br />
<br />
g*) x x 2 + x +1dx, x = 1 √ <br />
3sinht − 1<br />
2<br />
<br />
1<br />
.........................<br />
3 x 2 + x +1 <br />
3/2<br />
−<br />
1<br />
8 (2x +1)√ x 2 + x +1− 3 16 sinh−1 2x+1 √<br />
3<br />
+ c<br />
7. Per ciascuno degli integrali a) degli esercizi 1-6 precedenti, determinare la primitiva F (x) della<br />
funzione integranda che si annulla nel punto x 0 =1.<br />
8. Calcolare i seguenti integrali:<br />
<br />
x<br />
a) √ dx. ............................................................ 1<br />
1 − x<br />
4 2 arcsin x 2 + c <br />
<br />
x<br />
b) √ dx. .............................................. 2 √ x − 1+ 2<br />
x − 1<br />
3 (x − 1) √ x − 1+c <br />
<br />
1<br />
c) √ √ x (<br />
4<br />
x − 1) dx. ................................................[4 4√ x +4log| 4√ x − 1| + c]<br />
<br />
1<br />
d) √ dx. ............................................................. 1<br />
1 − 4x<br />
2 2<br />
arcsin 2x + c<br />
9x2<br />
e*) − 1dx. ............................................. √<br />
x<br />
2 9x2 − 1 − 1 6 cosh−1 (3x)+c <br />
<br />
1<br />
f)<br />
e x +1 dx. ............................................................[x − log (ex +1)+c]<br />
<br />
g) e x+ex dx. ........................................................................ e ex + c <br />
√ex<br />
h*) − 1dx. ........................................... 2 √ e x − 1 − 2arctan √ e x − 1 + c
4 M.GUIDA, S.ROLANDO<br />
i*)<br />
l)<br />
m)<br />
n)<br />
o)<br />
p)<br />
q)<br />
r)<br />
s)<br />
t)<br />
u*)<br />
<br />
1<br />
√<br />
e<br />
−2x<br />
− 1 dx..............................................................[arcsin (ex )+c]<br />
<br />
1<br />
dx...........................................................[log |1+logx| + c]<br />
x + x log x<br />
<br />
log x<br />
<br />
<br />
x 4+3log 2 x dx. ................................................... 1<br />
3<br />
4+3log 2 x + c<br />
log x<br />
√ dx. ............................................................. [2 √ x (log x − 2) + c]<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
sin 2 xdx. ................................................................ x−sin x cos x<br />
2<br />
+ c <br />
<br />
<br />
sin 3 cos<br />
xdx. ..............................................................<br />
3 x<br />
3<br />
− cos x + c<br />
1<br />
dx. ..............................................................[log |tan x| + c]<br />
sin x cos x<br />
1<br />
sin 2 dx. .........................................................[tan x − cot x + c]<br />
x cos 2 x<br />
cos x<br />
<br />
3 − cos 2 x dx. ...................................................... 1√ sin x<br />
2<br />
arctan √<br />
2<br />
+ c<br />
4sinx<br />
4cos 2 dx. .......................................[−2arctan(2cosx − 2) + c]<br />
x − 8cosx +5<br />
<br />
<br />
x<br />
x arctan (1 + 16x) dx..............<br />
2<br />
1<br />
2<br />
arctan (1 + 16x)+<br />
512 log 1+(1+16x) 2 <br />
− x 32 + c<br />
9. Per ciascuna delle seguenti funzioni definite a tratti (continue sul proprio dominio), calcolare tutte<br />
le primitive F (x) e determinare quella che vale 1 in x 0 =0:<br />
⎧<br />
⎨ x +2<br />
<br />
se x1<br />
<br />
log 1+25x<br />
2<br />
se x ≤ 1<br />
c) f (x) =<br />
x log 26 se x>1<br />
<br />
x log 1+25x<br />
2<br />
− 2x + 2 5<br />
arctan 5x + c se x ≤ 1<br />
...................... F (x) =<br />
log 26<br />
2<br />
x 2 + c se x>1 ,c=1<br />
⎧<br />
⎨ 4x +1 se x ≤ 0<br />
d) f (x) = 1<br />
⎩ √ se 0
<strong>INTEGRALI</strong> <strong>INDEFINITI</strong> 5<br />
ALTRE SOLUZIONI.<br />
Esercizio 7.<br />
a1) F (x) =log|x| + 9<br />
4x<br />
− 9 4 4<br />
a2) F (x) =e sin x − e sin 1<br />
a3) F (x) = 1 3 x3 log x − 3 1 +<br />
1<br />
9<br />
a4) F (x) =log|x − 2| − 4<br />
x−2 − 4<br />
a5) F (x) =x − 1 2 log e 2x +1 +arctane x − 1+ 1 2 log e 2 +1 − arctan e<br />
√<br />
a6) F (x) = x 2 x2 − 1 − 1 2 cosh−1 x