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6 ∫
4
4
3
dx.................................................. ... log |x|
dx...........................................................
....... x − 2
(^2) x
dx....................................................................................x x − 1
x
x log
3 x
3
cos^2 (5x
5
ANALISI MATEMATICA I - A.A. 2011/
INTEGRALI INDEFINITI / ESERCIZI
PROPOSTI
L’asterisco contrassegna gli esercizi più difficili.
- Calcolare i seguenti integrali usando la linearità dell’integrale:
∫. x^2 − 3
x^2 + 3
x^5 4 x ∫ x^2 − 4
3x^3 − 3
3 3 2
d) ∫
x dx......................................................................................[x − log |x + 1|
−
1 3 2
e) dx..........................................
x + x ∫ x^1 cos^2 x
g) ∫ tan^2 x dx......................................................................................[tan x − x
- Calcolare i seguenti integrali immediati usando la regola di integrazione per sostituzione:
a) ∫ esin^ x^ cos x dx..........................................................................................
esin^ x^ +
c
b) ∫ cos^5 x sin x dx...................................................................................Σ−
cos
x
cΣ
c)
cos (log x) dx...............................................................................[sin (log x)+
c] x
d) ∫ √
dx .........................................................................................
log x +
c
e)
√ (^) dx..........................................................................................3x + 1 + c
f) ∫
dx............................................................................
(^1) tan (5x + 9)
g) ∫ x^3 sin
x^4
dx .....................................................................................Σ−
cos x
+ cΣ
h) ∫ xe2x 2 −1dx........................................................................................... Σ 1 e2x 2 −1^ +
cΣ
i) ∫ x
1+ x^2 dx........................................................................
1+ x^2
1+ x^2
a )
b )
c )
f) 1+ sin
dx......................................................................................[x + cos x
3
dx.......................................................
..... .. 1 − x^3
1 − x^3
x^4
2
x
2
l )
m) ∫
x dx.................................................................................
arctan x^2
1
INTEGRALI INDEFINITI 3
2
(x^2 + 1)^2
2 1+x^2
i) ∫
dx...................................................
1 3 x 3 +5x (^) + 3
arctan x + c
(x + 1)
4 dx...........................................................................−^
(x+1)^3
c
dx. ......................................... ... x (^) e +1 + +
(ex^ − 2) (e^2 x^ + ex^ + 1)
dx........................... 2 log
(e
log
e + +1 +
h) ∫
dx..........................................................................
1 x
c
(x^2 + 1)^3 ∫ x
2
8 (x^2 +1)^2
Σ
8
1 3x^2 +3x+
- Calcolare i seguenti integrali mediante opportune sostituzioni: ∫ ex^ +1 Σ
e^2 x
2x
x
(5ex^ + 4) ex
x
2x x
l ) a ) b )
c) ∫ x
1 − xdx................................................
(^2) (1 − x) 2 √ 1 − x − 2 (1 − x) √ 1
− x + c
e) ∫
dx........................................................................................
log.^
1+sin x (^).
2 sin x + cos x − 1
2 tan(x/2)− 2
f) ∫
dx.................................................................Σ
1 log.
tan(x/2)
. +
cΣ
2 2
2 2
2 2
x^2
dx, x = sinh t..................................
.........
sin h
x c +
dx, t
x
x x^ x^ − 1
2 1 − x^4
4 x^2
(^38163)
2
2 6
4 2
x d) x ( 1 +
x)
dx .....................................[
5
x + log x − log (1 +
3
√ 3 x) −
arctan
x + c]
cos cos
cos^2 x g) 1 − 2 sin^2 x
dx............................................
.....
1
log
sin x+cos x sin x−cos x
1 x +
cΣ
h) ∫ tan^3 x dx...............................................................
tan^2 x − 1 log
1+ tan^2
x
- Calcolare i seguenti integrali mediante la sostituzione suggerita a fianco:
a) ∫
x^2 − 1 dx, x = cosh t................................................
x
x^2 − 1 −
1 cosh−1^ x
b) ∫
x^2 +1 dx, x = sinh t.............................................Σ
x
(1 + x^2 )+ 1 sinh−1^ x
+ cΣ
c*) ∫.
1 − x dx, x = sin t........................................................
arcsin x +
1 − x^2
1
x^2
−
√ x^2 +
x^2 − 1 √ 2
2
2
g*) x x^2 + x +1 dx, x = 3 sinh t − 1
x^2 + x + 1
1 ( 2 x + 1 )
x^2 + x + 1 −
3
sinh−^1.
2 x√+ 1
Σ + cΣ
- 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.
- Calcolare i seguenti integrali:
a)
x √ dx............................................................. arcsin x + c
b)
x √1) (^) x −dx. 1 +.... c.......................................... 2 x − 1 + (x −
x − 1 1 c) √ x (
x − 1 )
dx ............................................................[
3
x + 4 log
x − 1| + c]
d)
√ (^) dx......................................................................................arcsin 2x + c
e*) ∫
9x^2 − 1dx...................................................
Σx √ 9x^2 − 1 −
1 cosh−
(3x)+ c
d )
2
2
dx, x = sinh t.................................... (^) x (^2) +1 + ( + c
x +
e )
x −
f) −^1.^ ...^.....^.....^.....^....
..... ..
arctan
6
4
| Σ^ √
2
3
..^ Σ
4+3 log^2 x
3
x
dx ............................................................................................
x (log x − 2)
cos^2 x
2 2
2 512 32
(^2) xπ 2 − 21 π +^ c^ se^ x^ ≤
F (x)
5 log 26 x^2 + c se x >
, c = 4 x^ +^1 se^ x^ ≤
x^3
2
i*) ∫
dx.............................................................................[arcsin (ex)+
c] √ e−2x^ − 1 l)
dx......................................................... .. [log 1 + log x + c] x + x log x
m)
log x √ dx.................................................... 4 + 3 log x + c
∫ log x √
o) ∫ sin^2 x dx .......................................................................................
x−sin x cos x (^) +
c
p) ∫ sin^3 x dx......................................................................................Σ
cos
x − cos x +
cΣ
q)
dx............................................................ ... [log
tan x + c] sin x cos x
r)
dx.......................................................... [tan x
cot x + c] sin^2 x cos^2 x
s) ∫
cos x dx.......................................................
√^1 arctan
s√in x
t)
4 sin x dx........................................ [ 2 arctan (2 cos
x 2) + c] 4 cos^2 x − 8 cos x +
u*) ∫ x arctan (1 + 16x) dx. .............
x 2 arctan (1 + 16x)+ 1 log
16x)^2
− x^ + c
- 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 x < 0
x + 3 log |x − 1| + c se x < 0
a) f (x)
x − 1
F (x) =. 1 Σ e3x^
1 , c^ =^1 xe3x^ − 2 se x ≥ 0 . −x^3 sin
π + πx^2
se x
≤ 1
x − 3 3 − 2x + c + 9 se x ≥ 0
b) f (x)
x^2 − 8x + 7 se x > 1 Σ
sin(πx2)
2 co
s(πx2)
2 1 10 , c =
log 1+ 25x^2 se x 1 c) f (x) =
(^3) 4x − + 7x + 2π −^ c^ + 3 se^ x >^1
x log 26 se x > 1 Σ. x log
1+ 25x^2
− 2x + 2 arctan 5x + c se x
x
n )
F (x)
d) f (x) =
4 − x − 3 Σ. se 0^ < x^ ≤^4 2x^2 + x + c − 4 se x < 0
F (x)
4 − x − 6 log
4 − x
