Welcome to your Integration(Senior Class Math Study by Topics)

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1. Evaluate ∫ x / x+3 dx

x-3 + C

x - 3 In(x+3) + C

C

In(x + 3) + C

2. Find ∫sin 2x cos 3x dx

-1/10 sin5x + 1/2 cosx + C

-1/10 cos5x + 1/2 cosx + C

1/10 cos5x + 1/2 cosx + C

1/2 cosx + C

3. The line 2y = x + 2 intersects the curve y = 1/4 (7x-x²). Calculate the area of the finite region bounded by the curve and the line

6/7

9/8

7/8

5/6

4. Find ∫ 1/ x²+3x-10 dx

In (x-2 / x-5) + C

1/7 In (x-2 / x-5) + C

C

7In x + C

5. The velocity v m/s of a body moving at any time t, is given by v = 2t² - 1/3t³ + 10, determine the values of t at which the acceleration of the particle is zero.

0s, 4s

1/2s, 2.4s

-2s, 9s

3s, 5s

6. The gradient of a curve is given by dy/dx = 3x² - 8x + 3. If the curve passes through the origin, find the equation of the curve.

y = x³ - 4x² + 3x

y = x³ - 4x² - 3x

y = x³

y = x³ + 3x

7. Find the area of the finite regions included between the curve in a curve y with gradient dy/dx = 3x² - 6x + 2 and the x-axis

1/2

1/4

1/3

2

8. Find ∫ 1 / x²-7x+10 dx

In(x-5 / x-2) + C

1/3 In(x-5 / x-2) + C

C

In(x-5) + C

9. Given that dy/dx = 6x² + 5x, find the function which passes through the point (2, 30).

y = 6x² + 5x

y = x³ + 2x² + 4

y = 2x³ + 5/2 x² - 4

y = 2x³ + 5/2 x² + 4

10. Find in square units, the area of the finite region bounded by the curve y = x³ - x and the x-axis, from x = -1 to x =1

1/3

1/4

2

1/2

11. Given that x and t are two variables such that dx/dt = 3t² - 4t, and x = 1 when t = 4, express x in terms of t.

x = t³ - 2t² - 31

x = 3t³ - 2t² - 31

x = t³ - 2t - 31

x = 3t³ + 2t² - 31

12. Calculate the area of the finite region bounded by the curve y = 4x - x² and the x-axis

23/4

34/11

32/3

43/8

13. The gradient of the curve that passes through the point (1, -10) is given by 3x² - 6. Find the equation of the curve.

y = x³ + 5

y = x³ - 6x - 5

y = x³ + 6x - 5

y = x³ - 30

14. The acceleration of a particle moving in a straight line is given by a = 3t² - 4t. Find the expressions for the velocity given that v = 0 m/s and t = 0 s.

v = t³ + 2t²

v = 2t³ - t²

v = t³ - 2t²

v = t³ - 2t

15. An object travels in a straight line with velocity, v m/s at time t s is given by v = 3t² + 7. What is the distance travelled by the object from t = 2 s to t = 5 s?

138 m

168 m

112 m

144 m

16. Find ∫1 / x(x-1) dx

In (x/ x-1) + C

In (x-1) + C

In (x+1) + C

In (x/ x+1) + C

17. Find ∫x sin 2x dx

-xcos2x /2 + sin2x /2+ C

xcos2x /2 + sin2x /2 + C

cos2x /2 + C

sin2x /2 + C

18. The gradient of a curve is given by x² - 4x + 3 If the curve passes through the point (3, 1). Find the minimum point on the curve

(3, 1)

(1, 7)

(0, 6)

(2, 9)

19. The table below shows the velocity V m/s of a particle in a relation to time t s within a period of t seconds

t 0 1 2 3 4 5 6 V 10 12 15 16 11 5 3

Use trapezoid rule: 1/2 h(y1 +2y2 + 2y3 + ... + yn) Find the approximate distance traveled

63.5 sq units

60.5 sq. units

6.5 sq. units

65.5 sq. units

20. Evaluate the definite integral:

-4

5

4

3

21. Given that dv/dt = -64t and v = 10 when t = 0, find v when t = 2.

v = 118

v = -64

v = -118

v = 64

22. Find the area bounded by the curve y = -5 + 6x - x² and the x-axis

43

32/3

24.7

12

23. The gradient of a curve at P(x, y) is 3x². If the curve passes through Q(1, 5), find the equation of the curve.

x+4

x³+4

x³+4x

3x³+4

24. Find the area bounded by the curve y = x³ -3x² -x + 3 and the x-axis