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Calculus 12 Name _____________ LG 4 – 6 Worksheet Package Part A:

1. Find the slope, x-intercept and y-intercept of each of the following lines:

a)!!2x ! 4y+8 = 0!!!!!!!!!!!!!!!!!!!!!!!b)!!23x ! 14y = 2!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! x

2!y5= 4

2. Find the slope of a line parallel to and the slope of a line perpendicular to each of the lines in

question #1.

3. Find the equation of the line passing through:

a)!!(!1,!2)!with!slope!of !! 12!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!!(2,!!1)!and !parallel !to!3x ! 2y = !6!!!!!!!!!!!!!!!!!!!!!!!

!b)!!the!pts!(3,!1)!and !(!2,!!5)!!!!!!!!!!!!!!!!!!!!!!!d)!!(2,!!1)!and !perpendicular !to!3x ! 2y = !6!!!!!!!!

4. How can you check your answers to question #3 using a graphing calculator?

5. Use algebra and geometry to find the equation of the tangent line to the given circle at the given point. How can you check your answers using a graphing calculator? a)!!x2 + y2 = 25!!!!at !!(!3,!4)!!!!!!!!!!!!!!!!!!!!!!!!b)!!x2 + y2 =169!!!!at !!(!5,!!12)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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6. Use the first derivative to find the slope of the tangent line to the given curve at the given point:

a)!!y = 2x2 + 6!!!!at !!(!1,!8)!!!!!!!!!!!!!!!!!!!!b)!!y = !x2 + 2x !3!!!!at !!(2,!3)!

c)!!y = 4!3x3 !!!!at !!(1,!1)!!!!!!!!!!!!!!!!!!!!!!d)!!y = 3x !1x +3

!!!!at !!(!2,!!7)!

7. Find the slope of the normal line to the given curve at the given point for each of the curves in

question #6.

a) b) c) d)

8. Find the equation of the tangent line to the given curve at the given point for each of the curves in

question #6. Check answers using a graphing calculator.

a) b)

c) d)

9. Find the equation of the normal line to the given curve at the given point for each of the curves in question #6. Check answers using a graphing calculator.

a) b)

c) d)

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10. For each curve below find the equation of the (i) tangent line and (ii) normal line to the given

curve at the given point:

a)!!y = (x3 ! 5x + 2)(3x2 ! 2x)!!!!!!!at !!(1,!!2)!!!!!!!!!!!!!!!!!!!!!!b)!!y = 16x3 !!!!!!!!!!!!!at !!(4,!32)

c)!! x2

100!+! y

2

25!=!1!!!!!!!at !(!8,!!3)

11. Tangent lines are drawn to the parabola y = x2 !at !(2,!4)!and ! !18,! 164

"

#$

%

&'.

Prove that the tangents are perpendicular.

12. Find a point on the parabola y = !x2 +3x + 4 where the slope of the tangent line is 5.

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13. Find the equation of the normal line to the curve y = !x2 + 5x that has slope of -2.

14. Find the equations of the tangent lines to the curve y = 2x2 +3 that pass through the point (2, -7).

15. Prove the curve y = !2x3 + x ! 4 has no tangent with a slope of 2.

16. At what points on the curve y3 !3x = 5 is the slope of the tangent line equal to 1?

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Part B:

1. Use the CHAIN RULE to find dydx

at the indicated value x:

a)!!y = 2u2 + 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 3x !!!!!!!!!!!!!!!!!!!!!!!!!x =1

b)!!y = u2 ! 5u!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 2x +1!!!!!!!!!!!!!!!!!!!!x = 0

c)!!y = 5u+ 2

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 3x ! 2!!!!!!!!!!!!!!!!!!!!x =1

d)!!y = u2 +3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 2x2 !1!!!!!!!!!!!!!!!!!!!!x =1

e)!!y = 1!u1+u

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = x2 + 5 !!!!!!!!!!!!!!!!!!x = !2

2. Use the CHAIN RULE to find dydx

at the indicated value of x :

a)!!y = 2u2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 3v!!!!!!!!!!!!!!!!!!!!!!!!!v = 2x +1!!!!!!!!!!!!!!!!!!!!!!!!x = 0

b)!!y = 5+ 2u!!!!!!!!!!!!!!!!!!!!!!!!u = 2v!3!!!!!!!!!!!!!!!!!!!!v =1! 4x !!!!!!!!!!!!!!!!!!!!!!!x =1

c)!!y = 4u3 !3u2 !!!!!!!!!!!!!!!!!!!u = 2v2 + 4v!!!!!!!!!!!!!!!!!v =1! 2x2 !!!!!!!!!!!!!!!!!!!!!x = !1

d)!!y = 2+u !!!!!!!!!!!!!!!!!!!!!!!u = 2+ v !!!!!!!!!!!!!!!!!!!!!v = 2+ x !!!!!!!!!!!!!!!!!!!!x = 2

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Part C: 1. Use Calculus to find the CRITICAL POINTS of each of the following functions: a)!!!!y = x2 ! 6x + 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!y = 2x3 ! 24x !!!!!!!!!!!!!!!!!!!!!!c)!!y = 2x3 + 6x2 + 6x ! 2. Use Calculus showing a THUMBNAIL SKETCH to find the TURNING POINTS of each of the following functions. Check your answer using a graphing calculator. a)!!y = x2 ! 4x !3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!y = !2x2 + 6x +13

c)!!y = x3 !12x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!y = 2x3 + 9x2 +12x

e)!!y = x3 +3x2 +3x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!y = 3x4 ! 4x3

3. Find the local extrema (local maximum & minimum) using the FIRST DERIVATIVE TEST.

Show a THUMBNAIL SKETCH. a)!!y = 2x3 +3x2 !12x !!!!!!!!!!!!!!!!!!!b)!!y = 4x3 ! 48x +10!!!!!!!!!!!!!!!!!!!!c)!!y = 1! x2( )

2! 2!!

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Part D: 1. Find the second derivative of each of the following functions:

a)!!y = 3x5 ! 2x2 !!!!!!!!!!!!!!!!!!!!!!!!!!b)!!y = 6x2 !3x + 9!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!!y = 2x + 3x

d)!!y = 5x! 6x2 !!!!!!!!!!!!!!!!!!!!!!!!!e)!!y = 5

x!6x2+ 7!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!y = 1

x + 4

2. Use Calculus showing a THUMBNAIL SKETCH to find the INFLECTION POINTS of each of the following functions: a)!!!y = x3 !12x !!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!y = x3 +3x2 +3x !!!!!!!!!!!!!!!!!!!!!!!!c)!!y = x2 ! 4x !3 3. Use the SECOND DERIVATIVE TEST to find and classify all local extrema: a)!!!y = x2 !10x +3!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!y = x3 !12x + 5!!!!!!!!!!!!!!!!!!!!!!!!c)!!y = x4 ! 2x3

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Part E: 1. Find the GENERAL ANTIDERIVATIVE of each of the following functions: Verify your answers using differentiation.

a)!!!!4x3 !3x2 !!!!!!!!!!!!!!!!!!!!!!!!!!b)!!!!2x2 !8x + 6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!!!!3x5 + 4x3 ! 7!!!

d)!!!!x4 + 1x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!!!!x4 + 1

x2+ ln5!5x !!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!!! 3

x5+ 2x ! 7

g)!!!!10x+ xe !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!!!!x

23 ! x3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!!!2cos x !3x

j)!!!!cos(2x)+ 4x !!!!!!!!!!!!!!!!!!!!!!!k)!!!!3 x ! 1x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!l)!!!!10ex ! 2e2 x

m)!!!! 3x!!! 4

x2!+!2! !!!!!!!!!!!!!!!!!!!!!n)!!!! x

x2 +1!+!6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!o)!!!!x2 cos(x3)

p)!!!!2e! x + x!2 ! 5!!!!!!!!!!!!!!!!!!!!!!!!q)!!!! 5x2

x3 !1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!r)!!!!!e 3 ! x ! x2ex

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2. If:

a)!! dydx

= 7e!2 x !,!!then!y =

b)!! dydx

= 3x2 + 5x ! 2!,!!then!y =

c)!! dydT

= sinT + sin2T + sin3T !,!!then!y =

d)!! dydu

=1u!!! 1

u!1!,!!then!y =

e)!!dpdx

= ex + xe !,!!then!p =

f )!!dfdt= 5sec2 3t !,!!then! f =

3.!!!If !y = f (x) = x2 ! 6x + 5!!!!then! find :

!!!!a)!!!the!derivative!of !y!with!respect !to!x !!!!!!!!!!!!!!!!b)!!y '!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! dydx

d) the rate of change of y with respect to x e) the formula for the slope of the tangent line to f(x)

f )!!y '!at !x = 2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g)!! dydx!!at !!x = 2

h)!! f (2)!!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!! f '(2)!!!!!!!!!!!!!!!!!!!!!!!! j)!! f ''(2)!!!!!!!!!!!!!!!!!!!k)!!y '!when!y = 0

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4. If p = 4x2 !3x + 2 find:

a)!!the!general !antiderivative!of !p!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!y!if ! dydx!=!p

c)!!y!if !y ' = p!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!y!if !the! first !derivative!of !y = p

5. Find: a) the general antiderivative of x2 + x !8

b)!!if ! dydx

= x2 + x !8!!!!then!y =!!

c) all the solutions to the differential equation dydx

= x2 + x !8

d) the unique solution to dydx

= x2 + x !8 satisfying the given initial condition y(0) = 6

e) the equations of all curves y = f(x) whose tangent line has a slope of x2 + x !8 f) the rate of change/growth or the decay rate of change of y with respect to x is given by the expression x2 + x !8 . Find an expression for y. Part F: 1. Find the position function s(t) for an object with velocity function v(t): a)!!v(t) = 2t2 !3t !!!!!!!!!!!!!!!!!!!!!!!!!b)!!v(t) = t3 + 4t + 6!!!!!!!!!!!!!!!!!!!!!!!!!c)!!v(t) = t2 ! 5t 2. Find the position function s(t) for an object with velocity function v(t) and initial position s(0): a)!!v(t) = 3t ! t2,!!!s(0) = 5!!!!!!!!!!!!b)!!v(t) = 6t,!!!s(0) = 7!!!!!!!!!!!v(t) = t2 + 2t !,!!!!s(0) = 4

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3. Find the velocity function v(t) for an object with acceleration function a(t) and initial velocity v(0):

a)!!a(t) = 5,!!v(0) =10!!!!!!!!!!!!!!!!!!!b)!!a(t) = t !1,!!v(0) =1!!!!!!!!!!!!!!!!!!!!c)!!a(t) = t2 + t,!!v(0) = 0

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

4. Find the position function s(t) for an object with acceleration function a(t), initial velocity v(0) and initial position s(0).

a)!!a(t) = 5,!!v(0) =10!!!!!!!!!!!!!!!!!!!b)!!a(t) = t !1,!!v(0) =1!!!!!!!!!!!!!!!!!!!!c)!!a(t) = t2 + t,!!v(0) = 0

!!!!!!!!!!!!!!!!!!!!s(0) = 20!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!s(0) = 0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!s(0) = 0

Part G: 1. Find each of the following INDEFINITE INTEGRALS. Check your answers by differentiation.

a)!! x2 !dx! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! ! !dx! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! x"3 !dx!

d)!! 8x3 !dx! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! 1x "1

!dx! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! 11" x

!dx!

g)!! e4y !dy! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!! cos! y!dy! !!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!! ey + ye !dy!

j)!! e2 !dt !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!k)!! 5r !dr !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!l)!! sin2m!dm!!!

m)!! 6x2

1" x3!dx! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!n)!! v5 " 1

v4!dv!!!!!!!!!!!!!!!!!!!!!!!!!o)!! 5y!dr! !! !

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2. Find the EXACT VALUE of each of the following DEFINITE INTERGRALS. Check your answers using a graphing calculator.

a)!! 2x !dx1

3

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! x2 !dx"1

1

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! x " 6!dx"2

"1

!

d)!! x !dx1

2

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! ex !dx!

2!

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! x2 + 6!dx"2

"1

! !!

g)!! 3y!dy1

4

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!! 3y!dm1

4

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!! ln tt!dt

e

4

!

j)!! cost !dt0

!2

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!k)!! ex !dr2

5

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!l)!! xex2

!dm"3

"1

!

3. Simplify:

a)!!! 2x !dxa

b

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!! 3x2 !dxa

b

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!!! x2 !dx0

b

!

d)!!! 2x !dma

b

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!!! ep !dp0

b

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!! 2t!dt

!

4

!

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4. Find the exact area between the given curve and the x-axis over the given interval.

Check using a graphing calculator.

a)!!y = 2x,!!!!1! x ! 5!!!!!!!!!!!!!!!!!!!!!!!!b)!!y = x2,!!!!2 ! x ! 4!!!!!!!!!!!!!!!!!!!!!!!!!c)!!y = 2x3 + x,!!!!1! x ! 3

d)!!y = 3x,!!!!"1! x ! 5!!!!!!!!!!!!!!!!!!!!!!e)!!y = x2 " 4,!!!!"2 ! x ! 2!!!!!!!!!!!!!!!!!! f )!!y = 2x3,!!!!"1! x ! 4

5. Find the exact area between the given curve and the y-axis over the given interval.

Check using a graphing calculator.

a)!!y = 3x,!!!!0 ! x ! 4!!!!!!!!!!!!!!!!!!!!!!!!b)!!y = x2,!!!!0 ! x ! 3!!!!!!!!!!!!!!!!!!!!!c)!!y = 4x3,!!!!0 ! x ! 3

d)!!y = cos x,!!!!0 ! x ! !2!!!!!!!!!!!!!!!!!!!!e)!!y = ex,!!!!0 ! x ! 2!!!!!!!!!!!!!!!!!!!! f )!!y = "4x,!!!!0 ! x ! 4

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6. Find the exact area between the two curves f(x) and g(x) over the given interval. Check using a graphing calculator. a)!! f (x) = x2 !,!!!g(x) = x !,!!!!4 ! x ! 6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! f (x) = 4x !,!!!g(x) = "6x !,!!!!2 ! x ! 5

c)!! f (x) = "x2 " 2!,!!!g(x) = "x2 " 4!,!!!!"1! x ! 3!!!!!!!!!!!!!!!!!d)!! f (x) = ex !,!!!g(x) = 2x "3!,!!!!0 ! x ! 4 7. Find the exact area between the two curves f(x) and g(x) over the given interval. Check using a graphing calculator.

a)!! f (x) = 2!,!!!g(x) = !3!,!!!!!1" x " 4!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! f (x) = !2x +1!,!!!g(x) = !4x ! 5!,!!!!0 " x " 2

c)!! f (x) = ex !,!!!g(x) = 12x !,!!!!!2 " x " 0!!!!!!!!!!!!!!!!!!!!!!!!!d)!! f (x) = sin x !,!!!g(x) = !2x !,!!!!0 " x " !

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8. Find the exact area enclosed by the two curves f(x) and g(x). Check using a graphing calculator.

a)!! f (x) = x3 !!,!!!!g(x) = x2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! f (x) = 4x !!,!!!!g(x) = x2

c)!! f (x) = 2x2 !!,!!!!g(x) =12x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! f (x) = x3 !!,!!!!g(x) = x

9.!!a)!! find !! x3!2

3

" !dx !exactly.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! find !the!exact !area!under !the!curve

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!y = x3,! from!!2 # x # 3.

!!!!!c)!!why!are!the!answers!to!(a)!and !(b)!different?

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Part H: 1. Use ALGORITHM FOR EXTREME VALUES to find the maximum and minimum values of the given function on the given interval:

a)!! f (x) = x2 ! 4x +3,!!!!0 " x " 3!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! f (x) = x ! 2( )2 ,!!!!!0 " x " 2

c)!!g(x) = x3 !3x2,!!!!!1" x " 3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!h(x) = x3 !3x2,!!!!!2 " x "1!

e)!! j(x) = 2x3 !3x2 !12x +1!,!!!!2 " x " 0!

Part I: 1. Find the general antiderivative of each of each of the following functions:

a)!!x2 + x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!5x2ex3

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!!sin3 xcos x

d)!! 3x1+ x2

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!!3 x !!! 13 x

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!x3 +! 3 +! 3x3

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1. Cont’d: Find the general antiderivative of each of each of the following functions:

g)!!3sec2 2x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!!4sin xe2cos x !!!!!!!!!!!!!!!!!!!!!!!!i)!!5x + ex + e5 !! 4

j)!! 5x2

1! x3!!+! 5

x3!!!!!!!!!!!!!!!!!!!!!!!!!!k)!!4sec5x tan5x !!!!!!!!!!!!!!!!!!!!l)!!6x !!!6

x!!! 6

x2!+! 6

x3

2. Find each of the following: a)!! x2 + 7!dx !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! (x2 + 7)'!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! x + y!dy!!

d)!! 5x2 !dm!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! (e2 +! x5 )'!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! (xcos x2 )'!!!

3. Evaluate each of the following exactly:

a)!! e2 x !dx0

!

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! m!!dm" 2

5

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! yey2

!dy"!

!

!

d)!! 3y!dt" 2

!

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! 2y2 !!dx12

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! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! 4p+1

!dp0

e"1

!

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4. Simplify:

a)!! 2x !dxa

5

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! x3 !dx0

b

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! ex !dxa

0

!

d)!! yy3 +1

!dy4m

4m

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! y!dt2 p

4 p

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! c!dm"4a

6a

!

5. Solve for x:

a)!! 2m!dm!=!81

x

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! 4!dy!=!16x

1

! !!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! 2t +3!dt !=!40x

5

!

d)!! 4k !dy!=!8x2

x

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! x !dq!=!151

4

! !!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! r !dr !=!12

x

!

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6. Find the antiderivative by U-Substitution.

a)!! ! 3x " 5( )17 dx !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! ! 14x + 7( )6

!dx

!!!!!!!!u = 3x " 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 4x + 7

c)!! ! x x2 + 9 !dx !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! ! x2 2x3 " 4 !dx!!!!!!!!u = x2 + 9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u = 2x3 " 4

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e)!! !ln x( )10

x!dx !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! ! 5x

5+ 2x2!dx

g)!! ! 4xx2 +1

!dx !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!