Capítulo 6 - Integral Inde nida

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Capítulo 6 - Integral Inde…nida

1. Calcule as integrais inde…nidas abaixo usando integração imediata ou o método da substituição. (a) Z 1 + 1 x 3 dx (b) Z _dx (x + 2)3 (c) Z xpx 1dx (d) Z x3px2_{+ 3dx} (e) Z _ln3 x x dx (f) Z px x2 dx (g) Z psin x cos xdx (h) Z _sin x 3 2 + cos x₃ dx (i) Z p_x7 p3_x 6p4 _x dx (j) Z epx p x + 1 p xepx dx (k) Z _e1 x + 1 x2 dx (l) Z arctanpx (1 + x)pxdx (m) Z _secp x tgpx 2px dx (n) Z _e4x (1 + 3e4x₎2dx (o) Z _dx 1 + cos x (p) Z _{ln x} x + 4x ln2xdx (q) Z sinh (3x) dx (r) Z

tanh (ln (cos x)) tan xdx

2. Use o método de integração por partes para calcular as integrais inde…nidas abaixo. (a) Z x2cos xdx (b) Z x3 e2xdx (c) Z eaxcos (bx) dx_{, onde a e b 2 R} (d) Z ln x +p1 + x2 _dx (e) Z x sin (3x 2) dx (f) Z x sec2xdx (g) Z xnln 2x3 dx_{, n 2 N} (h) Z arcsin (2x) dx (i) Z arctan _p x 1 x2 dx (j) Z cos (ln x) dx 3. Resolva as integrais de funções trigonométricas abaixo.

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(a) Z sin4(ax) dx_{, a 2 R} (b) Z sin2x cos5xdx (c) Z sin2 x 2 cos 2 x 2 dx (d) Z sin2(3x) + cos(3x) 2dx (e) Z (2 sin (x))2dx (f) Z cot3(2x) csc (2x) dx (g) Z tan3(x) sec4(x) dx (h) Z cot3(2x) dx (i) Z tan2x sec3xdx

4. Calcule as integrais inde…nidas a seguir pelo método da substituição trigonométrica. (a) Z x 1 + x4dx (b) Z p 9 x2 x3 dx (c) Z px2 _a2_dx_{, a 2 R} (d) Z dx p x2 ₂₅ (e) Z _dx x 4 + ln2x (f) Z _{x + 4} 4x2_{+ 5}dx (g) Z dx a2_sin2_{x + b}2_cos2_x (h) Z dx (1 +px)32 (i) Z dx x3p_x2 ₄

5. Resolva as integrais inde…nidas elementares que contém um trinômio quadrado abaixo. (a) Z _{x + 3} x2 _2x ₅dx (b) Z (x 1)2 x2_{+ 3x + 4}dx (c) Z 5x + 3 p x2_{+ 4x + 10}dx (d) Z _x ₁ (x2_{+ 2x + 3)}2dx (e) Z 2x 1 (2x + x2₎p_{2x + x}2dx (f) Z _dx x2p_x2_{+ x + 2} (g) Z 2ex+ 1 ex_{+ 2} 5 ex dx (h) Z _x p 6x x2dx (i) Z _{cos x} sen2_x _{6 sin x + 12}dx

6. Use o método da decomposição em frações parciais para resolver as integrais in-de…nidas abaixo.

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(a) Z _5x ₂ x2 ₄dx (b) Z _x2 x2_{+ x} ₆dx (c) Z _4x ₂ x3 _x2 _2xdx (d) Z dx x3_{+ 3x}2 (e) Z x2 x 4 (x2_{+ 4) (2x} ₁₎dx (f) Z _x2 _{+ 2x} ₁ (x 1)2(x2_{+ 1)}dx (g) Z _3x2 _{x + 4} x (x3_{+ 2x}2_{+ 2x)}dx (h) Z x5+ 9x3+ 1 x3_{+ 9x} dx (i) Z 3e 2x+ 2e x 2 e 3x ₁ dx (j) Z 2x3 x + 1 x (x2_{+ 1)}2 dx (k) Z _x3_{+ 3x} ₁ (x 1) (x4 _{+ x}2₎dx

7. Use alguma das técnicas de integração estudadas para provar que: (a) Z _du u2_{+ a}2 = 1 aarctan u a + k (b) Z _du p u2 _a2 = ln u + p u2 _a2 _{+ k}

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(a) Z ln x x 1 + ln2x dx (b) Z sin x cos3_x+ x2 p 1 + x2 dx (c) Z _e2x ₁ e2x_{+ 1}dx (d) Z p 2 p3 _x 3 p x dx (e) Z sec2_x (tan2_{x + 9)}32 dx (f) Z

cos x sin xp1 sin4xdx (g) Z _p 1 + ex_dx (h) Z _ln2_x x + p x ln x dx (i) Z p x 1 + xdx (j) Z _dx p x 1 (k) Z dx p 1 + ex (l) Z x (ln x)2dx (m) Z _{ln (ln x)} x ln x dx (n) Z _dx 1 + e x (o) Z x2ln3(1 + x3) 1 + x3 dx (p) Z ln x p xdx (q) Z sin4 e2x cos e2x e2xdx (r) Z ₍p x + 1)13 p x dx (s) Z (ln (cos x))2tan xdx (t) Z dx 3 +p1 + 2x (u) Z ln x2+ 1 dx (v) Z _esec2_x tan x cos2_x dx (w) Z _ex p e2x_{+ 1}dx (x) Z x arcsin x p 1 x2 dx (y) Z _{ln [ln (ln x)]} x ln x dx (z) Z _{x tan}3p_x2 ₁ p x2 ₁ dx

9. Resolva as integrais inde…nidas abaixo pelo método que julgar conveniente. (a) Z _x p 1 + x p4 1 + xdx (b) Z ln (x + 3) p x + 3 dx (c) Z pcot (4x) p3 _{cot (4x)} p cot (4x) csc 2_{(4x) dx} (d) Z dx x2_csc8 2 x sec 3 2 x (e) Z tan (x) sec x cos2_{x + 9} dx (f) Z p ln2x 2 ln x + 10 x dx (g) Z sin (4x) ecos2xdx (h) Z _{(3 cos}2_x _{2) sin x}

cos3_x _cos2_{x + cos x} ₁dx

(i) Z e 2xcos2 3e x dx (j) Z 5x_{+ 9} 3x_{+ 1} 3x_{+ 9} x dx (k) Z 3x(sin (3x) + cos (3x))2dx (l) Z exp1 + ex_{ln (1 + e}x_{) dx} (m) Z p3 _x ₄ x 2p3x2_{+ 1} dx (n) Z tan3x cos2_xe 1+sec2x_dx

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Respostas: Ao resolver essas questões você poderá obter resultados equiva-lentes. 1. . (a) 1 2x2 6x 2_{ln x} _{6x + 2x}3 _{1 + k} (b) 1 2 (x + 2)2 + k (c) 2 15(3x + 2) (x 1) 3 2 + k (d) 1 5 x 2 ₂ _x2_{+ 3} 32 _{+ k} (e) ln 4_x 4 + k (f) x 1 ln + k (g) 2 3sin 3 2 x + k (h) 3 ln cos x 3 + 2 + k (i) 2x 3 4px7 51 2x1312 13 + k (j) 1 epx 2e 2px _{2 + k} (k) 1 x xe 1 x + 1 + k (l) arctan2(px) + k (m) sec(px) + k (n) 1 36e4x_{+ 12} + k (o) 1 cos x sin x + k (p) 1 8ln ln 2_{x +}1 4 + k (q) 1 3cosh (3x) + k (r) ln_{jcosh (ln (cos x))j + k} 2. .

(a) x2sin x 2 sin x + 2x cos x + k

(b) 1 8e 2x 4x3+ 6x2+ 6x + 3 + k (c) a (cos bx) e ax_{+ be}ax_{sin bx} a2_{+ b}2 + k (d) x ln x +px2_{+ 1} p_x2_{+ 1 + k} (e) 1 9sin (3x 2) 1 3x cos (3x 2) + k (f) x tan(x) + ln j cos(x)j + k (g) x n+1 (n + 1)2 (ln 2 + 3 ln x + n ln 2 + 3n ln x 3) + k (h) 1 2 p 1 4x2_{+ x arcsin(2x) + k} (i) p1 x2_{+ x arctan} _p x 1 x2 + k (j) 1 2x (cos (ln x) + sin (ln x)) + k 3. .

(a) sin(4ax) 8 sin(2ax) + 12ax

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(b) sin 7_x 7 2 sin5x 5 + sin3x 3 + k (c) x 8 sin(2x) 16 + k (d) 7x 8 + 2 sin3(3x) 9 + sin(6x) 48 sin3(3x) cos(3x) 12 + k (e) 9x 2 + 4 cos x sin(2x) 4 + k (f) csc(2x) 2 csc3_(2x) 6 + k (g) tan 6_x 6 + tan4_x 4 + k (h) cot 2_(2x) 4 ln(sin(2x)) 2 + k

(i) 2 tan x sec

3_x _{tan x sec x} _{ln(tan x + sec x)}

8 + k 4. . (a) arctan(x 2₎ 2 + k (b) 1 6 ln p 9 x2_{+ 3} x + 3p9 x2 x2 + k (c) x p x2 _a2 2 ln x +px2 _a2 _a2 2 + k (d) ln x +px2 _{25 + k} (e) 1 2arctan ln x 2 + k (f) 1 8ln x 2₊ 5 4 + 2p5 5 arctan 2p5x 5 ! + k (g) 1 abarctan a tan x b + k (h) 4( p x + 2) pp x + 1 + k (i) p x2 ₄ 8x2 1 16arctan 2 p x2 ₄ + k 5. . (a) ln (x 2 _2x ₅₎ 2 + p 6 3 ln p 6 x + 1 2 x2 _2x ₅ ! + k (b) x 5 ln (x 2_{+ 3x + 4)} 2 + 9p7 7 arctan 2x + 3 p 7 + k

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(c) 5px2_{+ 4x + 10} _{7 ln} x + 2 + p x2_{+ 4x + 10} p 6 ! + k (d) x 2 2(x2 _{+ 2x + 3)} 1 2p2arctan x + 1 p 2 + k (e) p3x + 1 x2_{+ 2x} + k (f) 4 p x2 _{+ x + 2} p_{2x ln(x + 4 +}p_8(x2_{+ x + 2)) +}p_{2 ln x} 8x + k (g) ln e2x+ 2ex 5 + p 6 12 ln ex₊p_{6 + 1} ex p_{6 + 1} ! + k (h) 3 arcsin 1 1 3x p x (x 6) + k (i) p 3 3 arctan p 3 sin x 3 p 3 ! + k 6. . (a) 2 ln (x 2) + 3 ln (x + 2) + k (b) x +4 ln (x 2) 5 9 ln (x + 3) 5 + k (c) ln x (x 2) 2 ln (x + 1) + k (d) (x ln x x ln (x + 3) + 3) 9x + k (e) 1 2ln x 2 + 4 1 2ln x 1 2 + k (f) ln (x 2_{+ 1)} _{2 arctan x + 2 ln (x} _{1) + 2x arctan x} _{2x ln (x} _{1) + x ln (x}2_{+ 1) + 2} 2 (1 x) + k (g) [7 x 14x arctan (x + 1) 5x ln (x 2_{+ 2x + 2) + 10x ln x + 8]} 4x + k (h) ln x 9 ln (x2+ 9) 18 + x3 3 + k (i) 3 ln (e 2x_{+ e}x_{+ 1)} 2 ln (e x _{1) +} p 3 3 arctan 2p3ex 3 + p 3 3 ! + k (j) 4 ln x 6x 2 ln (x 2_{+ 1) + 2 arctan x} _2x2_{ln (x}2_{+ 1)} _x2 _{+ 4x}2_{ln x + 2x}2_{arctan x} 4 (x2 _{+ 1)} + k (k) 6x ln (x 1) 6x arctan x + x ln (x 2_{+ 1)} _{8x ln x} ₄ 4x + k 7.

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8. . (a) 1 2ln ln 2 x + 1 + k (b) x p x2_{+ 1} _{ln x +}p_x2_{+ 1 + sec}2_(x) 2 + k (c) ln e2x+ 1 x + k (d) 6(2 3 p_x)5 2 5 4(2 3 p x)32 + k (e) tan(x) 9ptan2_{(x) + 9} + k (f) arcsin(sin 2_{x) + sin}2_xp₁ _sin4_x 4 + k (g) 2pex_{+ 1 + ln} p ex_{+ 1} ₁ p ex_{+ 1 + 1} + k (h) 2x 3 2 ln x 3 + ln3x 3 4x32 9 + k (i) 2px 2 arctan(px) + k (j) 2px + 2 ln_jpx 1_{j + k} (k) ln p ex_{+ 1} ₁ p ex_{+ 1 + 1} + k (l) x 2 _{2 ln}2 x 2 ln x + 1 4 + k (m) ln 2 (ln x) 2 + k (n) ln (ex+ 1) + k (o) ln 4 (x3+ 1) 12 + k (p) 2px (ln x 2) + k (q) sin 5_(e2x₎ 10 + k (r) 3( p x + 1)43 2 + k (s) ln 3_(cos(x)) 3 + k (t) p2x + 1 3 ln(3 +p2x + 1) + k (u) 2 arctan x 2x + x ln x2+ 1 + k (v) e 1 cos2 x 2 + k (w) ln(ex+pe2x_{+ 1) + k}

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(x) x arcsin(x)p1 x2_{+ k} (y) ln (ln x) [ln (ln (ln x)) 1] + k (z) tan 2₍p_x2 ₁₎ 2 + ln(cos( p x2 _{1)) + k} 9. . (a) 2(x + 1) 3 2 3 + 4(x + 1)54 5 + 4(x + 1)34 3 + x + k (b) 2 (ln (x + 3) 2)px + 3 + k (c) 6 cot 5 6(4x) 5 cot(4x) 20 + k (d) sin 11 2 x 22 sin9 _x2 18 + k (e) sec x 9 arctan(3 sec x) 27 + k (f) (ln x 1) p ln2x 2 ln x + 10 + 9 ln(ln x 1 +pln2x 2 ln x + 10) 2 + k

(g) ecos2x[2 cos(2x) 4] + k ou ecos2x 4 cos2x + 6 + k

(h) 5 ln (cos 2_{x + 1)} 4 5 arctan (cos x) 2 ln_{jcos x} 1_j 2 + k (i) e

2x_{[cos (6e} x_{) e}2x_{+ 6 sin (6e} x_{) e}x_{+ 18]}

72 + k (j) 27 2x ₆ x_{+ 2 arctan(3} x₎ 54 ln + k (k) 3 x_{+ sin}2₍₃x₎ ln 3 + k (l) 2 (e x_{+ 1)}32 _{[3 ln (e}x_{+ 1)} _2] 9 + k (m) 6 ln j1 + 2p3x2_j _{4 ln}_{jxj +} 3 arctan( p 2p3_x) p 2 + k (n) e tan2_x (tan2_x ₁₎ 2 + k