INTEGRALI INDEFINITI / ESERCIZI PROPOSTI

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