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MAT2455 - C´alculo Diferencial e Integral III para Engenharia Lista de Exerc´ıcios - Revis˜ao de integrais
Calcule as integrais indefinidas abaixo (de 1 a 73):
x^7 + x^2 + 1 x^2 dx 2.
e^2 x^ dx 3.
cos 7x dx
tg^2 x dx 5.
x − 2 dx 6.
tg^3 x sec^2 x dx
sen^3 x √ cos x dx 8.
tg x dx 9.
tg^3 x dx
x 1 + x^2 dx^ 11.
x 1 + x^4 dx^ 12.
x^2 1 + x^2 dx
x
1 − x^2 dx 14.
sec x dx 15.
x
1 + ln x
dx
x^2
x^3 + 1 dx 17.
4 x + 8 2 x^2 + 8x + 20 dx 18.
ln x x dx
dx (arcsen x)
1 − x^2
ex 1 + ex^ dx 21.
sen 2x 1 + cos^2 x dx
ex 3 x^2 dx 23.
ex^3
1 + ex^ dx 24.
sen
√ x x dx
earctgx 1 + x^2 dx 26.
2 x(x + 1)^2006 dx 27.
x sen x dx
ex^ cos x dx 29.
xr^ ln x dx, r ∈ IR 30.
(ln x)^2 dx
xe−x^ dx 32.
x arctg x dx 33.
arcsen x dx
sec^3 x dx 35.
cos^2 x dx 36.
sen^2 x cos^3 x dx
sen^2 x cos^2 x dx 38.
1 − sen x cos x dx 39.
3 x^2 + 4x + 5 (x − 1)(x − 2)(x − 3) dx
2 x^2 + 8x + 20 dx 41.
3 x^2 + 4x + 5 (x − 1)^2 (x − 2) dx 42.
x^5 + x + 1 x^3 − 8 dx
∫ (^) x 2 √ 1 − x^2
dx 44.
x^2
1 − x^2 dx 45.
e
√x dx
ln(x +
1 + x^2 ) dx 47.
√ dx 5 − 2 x + x^2
x ln x dx
sen(ln x) dx 50.
∫ (^) x x^2 − 4 dx 51.
∫ (^3) x (^2) + 5x + 4 x^3 + x^2 + x − 3 dx
a^2 + b^2 x^2 dx 53.
a^2 + b^2 x^2
dx 54.
x^2 − 2 x + 2 dx
3 − 2 x − x^2 dx 56.
(1 + x^2 )
1 − x^2
dx 57.
cos^3 x dx
sen^5 x dx 59.
cos^5 x sen^3 x dx 60.
sen^3
( (^) x 2
cos^5
( (^) x 2
dx
sen^5 x cos^3 x dx 62.
sen^4 x dx 63.
sen^2 x cos^5 x dx
sen^2 x cos^4 x dx 65.
cos^6 (3x) dx 66.
cos^2 x sen^6 x dx
sen^2 x cos^4 x dx 68.
∫ √^
1 − x 1 + x dx 69.
x − 3
x dx (Sugest˜ao: Fa¸ca u = 6
x)
x + 1 x^2 (x^2 + 4)^2 dx 71.
arctg x x^2 dx 72.
√x^2 dx 2 x − x^2
dx
∫ (^4) x (^2) − 3 x + 3 (x^2 − 2 + 2)(x + 1) dx
- Determine condi¸c˜oes sobre a, b, c, d ∈ IR para que as primitivas de
f (x) = (x^ −^ a)(x^ −^ b) (x − c)^2 (x − d)^2 sejam fun¸c˜oes racionais. (Resp.: d = c ou (a + b)(c + d) = 2(ab + cd))
- Calcule
x^2 (cos x + x sen x)^2 dx. Sugest˜ao: Calcule a derivada de v(x) = −^1 cos x + x sen x
. Use integra¸c˜ao por
partes. (Resp.: sen x − x cos x cos x + x sen x
+ C)
RESPOSTAS
- x
6 6
2 x 2
- sen 7x 7
- 7 ln |x − 2 | + k 6) tg^4 x 4
- 2
cos x
( (^) cos (^2) x 5
tg^2 x 2 + ln^ |^ cos^ x|^ +^ k^ 10)
2 ln(1 +^ x
(^2) ) + k
11)^1 2 arctg x^2 + k 12) x − arctg x + k
- − 1 3
(1 − x^2 )^3 + k 14) ln | sec x + tg x| + k
- 2
1 + ln x + k 16) 5 18
√ (^5) (x (^3) + 1) (^6) + k
- ln(2x^2 + 8x + 20) + k 18)^2 3
(ln x)^3 + k
ln |arcsen x| + k 20) ln(1 + ex) + k
− ln(1 + cos^2 x) + k 22)^1 3 ex 3
23)^3 4
√ (^3) (1 + ex) (^4) + k 24) −2 cos √x + k
- earctgx^ + k 26) 2(x + 1)^2005
( (^) x + 1 2006
- −x cos x + sen x + k 28)^1 2 ex(sen x + cos x) + k
xr+ r + 1 ln x − xr+ (r + 1)^2
- k se r 6 = − 1 1 2 (ln^ x)
(^2) + k se r = − 1
x(ln x)^2 − 2(x ln x − x) + k
(−x − 1)e−x^ + k 32) x^2 2 arctg x − x 2
arctg x + k
- x arcsen x +
1 − x^2 + k 34)
sec xtg x +
ln | sec x + tg x| + k
(x + sen x cos x) + k 36) sen^3 x 3
sen^5 x 5
x − sen 4x 4
