REVIEW 3

1. Calculate the antiderivative F(x) of the function y = f(x) such that

a)
, and F(0) = 1; b)
, and F(1) = 0;

2. Find the antiderivatives (straightforward)

a)
b)
; c)
.

3. Integrate by parts:

a)
; b)
; c)

4. Integrate by substitution:

a)
; b)
; c)

d)
; e)
, f)
.

5. a) Without finding the values of the coefficients express the following rational functions as sums of partial fractions:
,
,
,
,
,
,

b) find the antiderivatives of the following rational functions

6. Find the antiderivatives of the following rational functions (more complex ones)



,

7. Find the antiderivatives of

a)
[parts]; b)
; [ hint: t = sinx, then rational f-ction].

8. Find:

a)
, [hint: t = cosx] b)
, [hint: t = sinx] c), c)
, [hint: t = 3+2e^x], d)
[ hint: see below

.

9. Find the following definite integrals

a.
[rational] b.
[rational] c.
[subst t=x⁴] e.
[subst] g.
[parts] h.
[subst t=1+e^x]

i.
[parts] j.
[subs + parts] k.
[parts] l.
[subst: t=cosx] m.
[subst: t=cosx & rational ]

n.
[parts]

10. Find the area between the following curves:

1. 
   ,
   ; (b)
   ,
   .

11. Calculate the volume of the solid of revolution obtained by rotating around the
axis

(a) the curve
, x∈<0,π>;

(b) the figure bounded by
,
.

12. Calculate the surface area of the solid of revolution obtained by rotating around the
axis :

1. the curve
   , x∈[0,2];

2. the figure bounded by
   , y = x .

13. Find the arc length of the curve
, y>0 , x∈[0,1];

14. Find the arc length of the curve

for

15. Find the solutions of the `separable variables' differential equations:

16. For the following differential equations find a solution satisfying the given boundary condition.

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