Cálculo

Cálculo 3

1)∫5x dx=

∫〖5x〗^(1+1)/(1+1)+ c

((5x^2)/2+c)

dx (〖5x〗^2/2)+ c dx (〖5x〗^(2-1)/(2-1))+C

( (2-1).〖5x〗^(2-1)+0)/(2-1) = 〖5x〗^(2-1) = 5x

2) ∫(t^2+1/t^2 )dt =

∫t^2 dt+ ∫1/t^2 dt

∫t^2 dt+∫t^(-2) dt

t^(2+1)/(2+1)+t^(-2+1)/(-2+1)+c

t^3/3+t^(-1)/(-1)+c

dx (t^3/3+ t^(-1)/(-1))+ c

dx (t^(3-1)/(3-1)+ t^(-1-1)/(-1-1))+ c

((3-1) . t^(3-1))/(3-1)+ ((-1-1) .〖 t〗^(-1-1))/(-1-1)+0 t^2+ t^(-2) = t^2+ 1/t^2

3) ∫x^3 dx

∫x^(3+1)/(3+1)+c

∫x^4/4+c

dx(x^4/4+c)

dx(x^(4-1)/(4-1))+c

((4-1).x^(4-1))/(4-1)+0=

= x^3

4) ∫(x^2-2) dx =

∫x^2 dx - ∫2dx =

∫x^2 dx-2∫dx =

x^(2+1)/(2+1)- 2x+c

x^3/3-2x+c

dx (x^3/3)- 2x+c

dx (x^(3-1)/(3-1))- 2x+c

((3-1).x^(3-1) )/(3-1)- 2+0

x^(-2)-2

5) ∫(4t + 7) dt

∫4t dt + ∫7dt

4∫▒〖t dt +7∫dt〗

4 t^(1+1)/(1+1)+ 7x+c

4t^2/2+ 7x+c

dx (〖4t〗^2/2)+ 7x+c

dx (〖4t〗^(2-1)/(2-1))+ 7x+c

((2-1).4 t^(2-1))/(2-1)+7+0

4t+7

6) ∫ 4/t^2 dt

∫▒〖4 .t^(-2) dt〗

4∫t^(-2) dt

4∫t^(-2+1)/(-2+1)+ c

(4 t^(-1))/(-1)+ c

dt (〖4t〗^(-1)/(-1))+ 0

dt (〖4t〗^(-1-1)/(-1-1))+ 0

((-1-1).〖4t〗^(-1-1))/(-1-1)+0〖=4t〗^(-2) = 4/t^2

7) ∫ (x+1/√x)dx

∫xdx+ ∫1.x^(1⁄2) dx dx ( x^2/2+ 1^(3⁄2)/(3⁄2))+0

∫xdx+1 ∫x^(1⁄2) dx dx ( x^(2-1)/(2-1)+ 1^(3⁄2-1)/(3⁄2-1))

x^(1+1)/(1+1)+ 〖1x〗^(1/2+1)/(1/2+1)+ c ((2-1). x^(2-1))/(2-1)+ ( 3⁄2-1 .1^(3⁄2-1))/(3⁄2-1)

x^2/2+ 1^(3⁄2)/(3⁄2)+ c x+ 1^(1⁄2) = x 1/√x

x^2/2+ √x/(3⁄2)+ c

8) ∫(x^2+ 5x + 8) dx

∫x^2 dx+5∫xdx +8∫dx

x^(2+1)/(2+1)+ 〖5x〗^(1+1)/(1+1)+ 8x+c

x^3/3+ 〖5x〗^2/2+ 8x+c

dx (x^3/3)+(〖5x〗^2/2)+ 8x+0

dx (x^(3-1)/(3-1))+(〖5x〗^(2-1)/(2-1))+ 8+0

((3-1).x^(3-1))/((3-1)) + ((2-1) 〖5x〗^(2-1))/(2-1) + 8

x^2+ 5x+8

9) x . √x dx

∫xdx . ∫x^(1⁄2) dx

x^(1+1)/(1+1) x^(1⁄2+1)/(1⁄2+1)+c

x^2/2 x^(3⁄2)/(3⁄2)+ c

x^2/2 . √x/(3⁄2)+ c

dx(x^2/2 . x^(3⁄2)/(3⁄2))+ 0

((2-1) .x^(2-1))/(2-1) . ((3⁄2-1) . x^(3⁄2-1))/(3⁄2-1)

x .x^(1⁄2) = x .√x

10) ∫〖2x〗^7dx

2∫x^7 dx

2 x^(7+1)/(7+1)+ c

〖2x〗^8/8+ c

dx(〖2x〗^8/8)+ c

dx(〖2x〗^(8-1)/(8-1))+ c= ((8-1) .2x^(8-1))/(8-1)+ 0 = 〖2x〗^7

11)∫ 3/x^5 dx

∫▒〖3 .x^(-5) dx〗

3∫x^(-5) dx

3∫x^(-5+1)/(-5+1)+ c

(3 x^(-4))/(-4)+ c

dx (〖3x〗^(-4)/(-4))+ 0

dx (〖3x〗^(-4-1)/(-4-1))+ 0

((-4-1).〖3x〗^(-4-1))/(-4-1)+0〖=3x〗^(-5) = 3/x^5

12) ∫〖10〗^3 √(x^2 ) dx

〖10〗^3∫√(x^2 ) dx

〖10〗^3∫x^(2⁄2) dx

(〖10〗^3 x^(2/2+1))/2^(2⁄2+1) + c

(〖10〗^3 x^(4/2))/(4/2)+ c

〖10〗^3. (2/4) x^(4⁄2)+c

〖10〗^3 .(2/4).√(x^4 )+ c

13) ∫3/√x dx

3∫x^((-1)⁄2) dx

3 x^((-1)/2+1)/((-1)/2+1)+c

(3