5.4 Indefinite Integrals and the Net Change Theorem

1-4. Verify by differentiation that the formula is correct.

1. .

Let . Then .

That is

2. .

Let and . Then and by differentiation.

That is

3. .

Let then .

So

Also change of variable for , then .

That is

4. .

Let then .

So

.

Also change of variable for , then . That is

5-13. Find the general indefinite integral.

5. .

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6. .

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7. .

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8. .

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9. .

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10. .

11. .

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12. .

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13. .

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Let . Then .

So, .

14-22. Evaluate the integral.

14. .

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15. .

16. .

17. .

18. .

19. .

20. .

http://matrix.skku.ac.kr/cal-lab/cal-5-4-20.html

-1/2*sqrt(2)*log(-2*sqrt(6) + 5) + sqrt(2)*sqrt(6)

21. .

22. .

Let . Then

Thus, .

23. Estimate the area of the region that lies under the curve and above the -axis.

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0.876956449003

24. Water is being added to a tank at a rate of per minute. How much water is added to the tank from to ?

The amount of water added for is where

so that .

25. Find the area of the region that lies to the right of the -axis and to the left of the parabola .

Since the parabola meets the -axis at and , the area is given by

.

26. A particle is moving along a line with the velocity function . Find the dis-placement and the distance traveled by the particle during the time .

The displacement = , the distance .

27. A honeybee population starts with 30 bees and increases at a rate of bees per week. How many honeybees are there after 10 weeks?

Since the net change in population during 10 weeks is , the total number of honeybees after 10 weeks is .

28. The acceleration function (in ) of a particle is given by and the initial velocity is .

Find the velocity of the particle at time and determine the total distance traveled for .

Since , the velocity at time is . Therefore the distance traveled for is .