Integrais

Primeira Lista de Exercícios

ʃ5x dx= 5ʃx dx = 5.x²/2 + C Dx 5. x²/2 + C = 10 x/2 + C = 5x

ʃ x³ dx= ʃ x4/4 Dx = 4. x³/4 + C = x3

ʃ( x² + 5x + 8)dx = ʃx²dx + 5ʃx dx + 8ʃ dx = x³/3 + 5 x²/2 +8x + C Dx= 3 x²/3 + 10 x/2 + 8 = x² + 5x + 8

ʃ(t² + 1/t² ) = ʃt² dt + ʃ 1/t² dt= ʃt² dt + ʃ t-2 dt =

(t³ )/3 + (t-1)/(-1) + c = Dt = 3 (t² )/3 + (-1(t-2)/(-1) ) + c = t² + t-2 + c = T² + 1/t²

E) ʃ(4x + 7)dx = ʃ4x dx + ʃ7 dx = 4ʃx dx + 7ʃdx = 4(x² )/2 + 7x + c 2x² + 7x + c Dx = 4x + 7

F) ʃ(4.√x)dx = ʃ4x1/2 = 4ʃ dx = 4 (x 3/2)/(3/2) + c = 2/3.4x3/2 + c

Dx 3/2.2/3 4x1/2 + c = 4x1⁄2 + c = 4.√x

G) ʃ (T√T + 1/(T√T)) dt= ʃt.t1⁄2dt + ʃ t. t(-1)⁄2 dt = ʃt3⁄2 dt + ʃt(-3)⁄2 dt = (t 5⁄2)/(5/2) + (t (-1)⁄2)/(-1/2) + c = 2/5.t5/( 2) + 2/1.t (-1)/2 + c

Dt 5/2.2/5 t 3⁄2 + 1/2.2/1 t (-3)⁄2 + c = t3⁄2 + t(-3)⁄2 + c = √t³+1/√t³ + c = √(t².t) + 1/√(t².t) + c = T√T + 1/(T√T)

H) ʃ(x² - 2 )dx= ʃx² dx - ʃ2 dx = ʃx² - 2ʃdx = x³/3 - 2x + c

Dx 3x²/3 - 2 + c = x² - 2

I) ʃ4/x² dx = ʃ4.x-² dx= 4ʃx-² dx = 4(x-¹)/(-1) + c =

Dx (-1).4(x-²)/(-1) + c = 4 x-² + c = 4/x²

J) ʃ(x + 1/√x)dx= ʃx dx + ʃx (-1)⁄2 dx = x²/2 + (x1/2)/(1\2) + c = x²/2 +2/1 x1⁄2 + c

Dx 2 x/2 + 1/2.2/(1 ) x (-1)/2 + c = x + x(-1)/2 + c = x + 1/√x

Segunda Lista de exercícios

1) ʃ2x7 dx= 2ʃx7 dx = 2x8/8 + c = x8/4 + c=

Dx 8x7/4 + c = 2x7

2) ʃ3/x5 dx = ʃ3.x5 dx = 3ʃx5 dx = 3x6/6 + c = x6/2 + c

Dx 6 x5/2 + c = 3x5 + c = 3/x5

3) ʃ 10∛x2 dx= 10ʃ x2⁄3 dx = 10x(x 5⁄3)/(5\3) + c = 3/5 10x 5⁄3 + c

Dx 5/3.3/510x2⁄3 + c = 10x2⁄3 + c = 10∛x2

4) ʃ3/√x dx= ʃ3.x(-1)⁄2 dx 3ʃ x(-1)⁄2 dx= 3(x 1⁄2)/(1\2) + c = 2/(1 ) 3x1⁄2 + c =

Dx 1/2.2/13x (-1)⁄2 + c = 3 x(-1)⁄2 = 3/√x

5) ʃ7x³ √x dx = ʃ7x³. x1⁄2 dx = ʃ7x7⁄2 dx = 7ʃx7⁄2 dx = 7 (x 9⁄2)/(9\2) + c =

2/9 7 x9⁄2 + c

Dx 9/2.2/9 7 x7⁄2 + c = 7x7⁄2 + c = 7√(x².x².x².x) + c= 7x³ √x

6) ʃ(3x² - 2x³) dx= ʃ3x² dx - ʃ 2x³ dx = 3ʃx² dx – 2ʃx³ dx= 3x³/3 - 2x4/4 + c = x³ - x4/2 + c

Dx 3x² - 4x³/2 + c = 3x² - 2x³

7) ʃ x4( 5 - x² )dx= ʃ 5x4 – x6dx = 5ʃx4 dx - ʃ x6 dx = 5x5/5 – x7/7 + c

X5 – x7/7 + c

Dx 5x4 – 7 x6/7 + c = 5x4 – x6 = 54(5 – x²)

8) ʃ( 4x³ - 3x² + 6x -1) dx= 4ʃx³dx - 3ʃx²dx + 6ʃx dx - ʃdx=

4x4/4 - 3x³/3 +6x²/2 - x + c = x4 – x³ +3x² - x + c

Dx 4x³ - 3x² + 6x - 1

Terceira lista de exercícios

ʃ(5 cos x – 4 sen x ) dx = 5ʃcos dx – 4ʃsen x dx =

5sen x + cos x + c

ʃ (sen x)/cos²x dx= ʃ (sen x)/cos⁡〖x .cos⁡x 〗 dx = ʃ1/cos⁡x . (sen x)/cos⁡x dx

ʃsec x . tg x dx = sec x + c