INTEGRALI INDEFINITI / ESERCIZI PROPOSTI

ANALISI MATEMATICA I - A.A. 2011/2012
INTEGRALI INDEFINITI / ESERCIZI PROPOSTI
L’asterisco contrassegna gli esercizi più difficili.
1. Calcolare i seguenti integrali usando la linearità dell’integrale:
x 2 − 3 x 2 +3
a)
x 5 dx..................................................... log |x| + 9
4x
+ c
4
x 2 − 4
b)
x − 2 dx. ................................................................. 1
2 x2 +2x + c
3x 3 − 3
c)
x − 1 dx. ........................................................... x 3 + 3 2 x2 +3x + c
x
d) dx. ..............................................................[x − log |x +1| + c]
1+x
x 3 + x 2 +1
e)
dx. ......................................... 1
x − 1
3 x3 + x 2 +2x +3log|x − 1| + c
cos 2 x
f)
dx. ................................................................[x +cosx + c]
1+sinx
g) tan 2 xdx. ..................................................................[tan x − x + c]
2. Calcolare i seguenti integrali immediati usando la regola di integrazione per sostituzione:
a) e sin x cos xdx. .................................................................. e sin x + c
b) cos 5 x sin xdx. ............................................................... − cos6 x
6
+ c
cos (log x)
c)
dx. ..............................................................[sin (log x)+c]
x
1
d)
x √ log x dx. ................................................................. 2 √ log x + c
1
e) √ dx................................................................ √
2
3x +1
3 3x +1+c
1
f)
cos 2 (5x +9) dx. ...................................................... 1
5
tan (5x +9)+c
g) x 3 sin x 4
cos x4
dx. ............................................................... −
4
+ c
h) xe 2x2−1 1
dx. .................................................................
−1 4 e2x2 + c
i) x 1+x 2 dx...................................................... √ 1
3 1+x
2
1+x 2 + c
x 2
l) √ dx. ............................................................. √
− 2
1 − x
3 3 1 − x3 + c
x
m)
1+x 4 dx. ............................................................... 1
2 arctan x2 + c
1

2 M.GUIDA, S.ROLANDO
3. Calcolare i seguenti integrali usando la regola di integrazione per parti:
a) x 2 log xdx. ..........................................................
1
3 x3 log x − 1 3 + c
b) x 2 cos xdx. .................................................. x 2 − 2 sin x +2x cos x + c
x
c) x 2 e 2x
dx. ........................................................... 2 − x + 1 e
2x
2 2 + c
d) x cosh (3x) dx. ................................................ 1
3 x sinh 3x − 1 9
cosh 3x + c
x
e) (x +5)logxdx. ...........................................
2
2 +5x log x − x2
4 − 5x + c
log x
f)
x 2 dx. ................................................................... − 1+log x
x
+ c
g) arctan xdx. ............................................... x arctan x − 1 2 log 1+x 2 + c
h) arcsin xdx. ..................................................... x arcsin x + √ 1 − x 2 + c
i) x cos 2 1 x sin 2x cos 2x
xdx. .................................................
2 2
+
4
+ x2
2
+ c
l) e 2x cos xdx. ........................................................... sin x+2 cos x
5
e 2x + c
4. Calcolare i seguenti integrali di funzioni razionali:
x +2
a)
x 2 − 4x +4 dx. .................................................... log |x − 2| − 4
x−2 + c
2x +5
b)
x 2 +2x − 3 dx. ........................................... 7
4 log |x − 1| + 1 4
log |x +3| + c
x +1
c)
x 2 − 2x +4 dx. .................................. 1
2 log(x2 − 2x +4)+ √ 2
3
arctan x−1 √
3
+ c
x +1
d)
x 2 − x +5 dx. ................................... 1
2 log(x2 − x +5)+ √ 3
19
arctan 2x−1 √
19
+ c
2x 2 + x
e)
(x +2)(x 2 +2x +6) dx. ............ log |x +2| + 1 2 log x 2 +2x +6
− √ 4
5
arctan x+1 √
5
+ c
3x − 1
f)
(x − 1) (x − 2) 2 dx. ................................................ 2log x−1
x−2
− 5
x−2 + c
x 4 − 5x 3 +8x 2 − 9x +11
x
g)
x 2 dx. .....................
3
− 5x +6
3 +2x − log |x − 2| +2log|x − 3| + c
1
h)
(x 2 +1) 2 dx. .................................................... 1 x
2 1+x
+ 1 2 2 arctan x + c
1
i)
(x 2 +1) 3 dx. .................................................. 1 3x 3 +5x
8
+ 3 (x 2 +1) 2 8 arctan x + c
x 2
l)
(x +1) 4 dx. ............................................................ − 1 3x 2 +3x+1
3
+ c
(x+1) 3
5. Calcolare i seguenti integrali mediante opportune sostituzioni:
e x +1
a)
e 2x +1 dx............................................. x − 1 2 log e 2x +1 +arctane x + c
(5e x +4)e x
b)
(e x − 2) (e 2x + e x +1) dx. .......................... 2log(e x − 2) − log e 2x + e x +1 + c

c)
d)
e)
f)
g)
h)
x √ 1 − xdx. ....................................
INTEGRALI INDEFINITI 3
2
5 (1 − √ x)2 1 − x − 2 3 (1 − x) √
1 − x + c
1+ √ x
x (1 + 3√ x) dx. ...............................[6 6√ x +logx − log (1 + 3√ x) − arctan 6√ x + c]
1
cos x dx. ................................................................ log 1+sin x cos x
+ c
1
2sinx +cosx − 1 dx................................................ 1
2 log tan(x/2)
tan(x/2)−2
+ c
cos 2 x
1 − 2sin 2 x dx. ................................................ 1
4 log sin x+cos x
sin x−cos x + 1 2 x + c
tan 3 xdx. ............................................... 1
2 tan2 x − 1 2 log 1+tan 2 x + c
6. Calcolare i seguenti integrali mediante la sostituzione suggerita a fianco:
x2
a) − 1 dx, x =cosht...................................... √
x
2 x2 − 1 − 1 2 cosh−1 x + c
x2
x
b) +1dx, x =sinht....................................
2 (1 + x2 )+ 1 2 sinh−1 x + c
1 − x
c*)
1+x dx, x =sint. ........................................... arcsin x + √ 1 − x 2 + c
1
d)
(x 2 +1) √ x 2 +1 dx, x =sinht. ............................................. √ 1
+ c x 2 +1
√
x2 +1
e)
x 2 dx, x =sinht. .......................................... sinh −1 x − √
x 2 +1
x
+ c
√
x2 − 1
f)
dx, t = x
x
2 − 1. ............................. √ x 2 − 1 − arctan √ x 2 − 1 + c
g*) x x 2 + x +1dx, x = 1 √
3sinht − 1
2
1
.........................
3 x 2 + x +1
3/2
−
1
8 (2x +1)√ x 2 + x +1− 3 16 sinh−1 2x+1 √
3
+ c
7. Per ciascuno degli integrali a) degli esercizi 1-6 precedenti, determinare la primitiva F (x) della
funzione integranda che si annulla nel punto x 0 =1.
8. Calcolare i seguenti integrali:
x
a) √ dx. ............................................................ 1
1 − x
4 2 arcsin x 2 + c
x
b) √ dx. .............................................. 2 √ x − 1+ 2
x − 1
3 (x − 1) √ x − 1+c
1
c) √ √ x (
4
x − 1) dx. ................................................[4 4√ x +4log| 4√ x − 1| + c]
1
d) √ dx. ............................................................. 1
1 − 4x
2 2
arcsin 2x + c
9x2
e*) − 1dx. ............................................. √
x
2 9x2 − 1 − 1 6 cosh−1 (3x)+c
1
f)
e x +1 dx. ............................................................[x − log (ex +1)+c]
g) e x+ex dx. ........................................................................ e ex + c
√ex
h*) − 1dx. ........................................... 2 √ e x − 1 − 2arctan √ e x − 1 + c

4 M.GUIDA, S.ROLANDO
i*)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u*)
1
√
e
−2x
− 1 dx..............................................................[arcsin (ex )+c]
1
dx...........................................................[log |1+logx| + c]
x + x log x
log x
x 4+3log 2 x dx. ................................................... 1
3
4+3log 2 x + c
log x
√ dx. ............................................................. [2 √ x (log x − 2) + c]
x
sin 2 xdx. ................................................................ x−sin x cos x
2
+ c
sin 3 cos
xdx. ..............................................................
3 x
3
− cos x + c
1
dx. ..............................................................[log |tan x| + c]
sin x cos x
1
sin 2 dx. .........................................................[tan x − cot x + c]
x cos 2 x
cos x
3 − cos 2 x dx. ...................................................... 1√ sin x
2
arctan √
2
+ c
4sinx
4cos 2 dx. .......................................[−2arctan(2cosx − 2) + c]
x − 8cosx +5
x
x arctan (1 + 16x) dx..............
2
1
2
arctan (1 + 16x)+
512 log 1+(1+16x) 2
− x 32 + c
9. Per ciascuna delle seguenti funzioni definite a tratti (continue sul proprio dominio), calcolare tutte
le primitive F (x) e determinare quella che vale 1 in x 0 =0:
⎧
⎨ x +2
se x1
log 1+25x
2
se x ≤ 1
c) f (x) =
x log 26 se x>1
x log 1+25x
2
− 2x + 2 5
arctan 5x + c se x ≤ 1
...................... F (x) =
log 26
2
x 2 + c se x>1 ,c=1
⎧
⎨ 4x +1 se x ≤ 0
d) f (x) = 1
⎩ √ se 0

INTEGRALI INDEFINITI 5
ALTRE SOLUZIONI.
Esercizio 7.
a1) F (x) =log|x| + 9
4x
− 9 4 4
a2) F (x) =e sin x − e sin 1
a3) F (x) = 1 3 x3 log x − 3 1 +
1
9
a4) F (x) =log|x − 2| − 4
x−2 − 4
a5) F (x) =x − 1 2 log e 2x +1 +arctane x − 1+ 1 2 log e 2 +1 − arctan e
√
a6) F (x) = x 2 x2 − 1 − 1 2 cosh−1 x