# Exercises 9. Fundamental theorem of Calculus (antiderivative part

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```					                               Razvan SATNOIANU                                              Calculus 106     2004

Exercises 9. Fundamental theorem of Calculus
(antiderivative part). Fundamental theorem of
Calculus (definite integral). Improper integrals.

Fundamental Theorem of Calculus (The antiderivative part)

1. Find the derivative of each of the following functions:
x                                 4                         9x

∫       7 + 6t dt ; b) ∫ 3 tan t dt ; c)                    ∫ cos(t
3
a)                                                                              ) dt .
1                                 x                       cos( x )

x
1
2. Find the interval on which the function                                 ∫ 2 + 7t dt
0
is convex.

Fundamental Theorem of Calculus (Definite integrals). Improper integrals.
Further applications

3. Evaluate the integrals:
π /6                             π /4                                       1
∫ (sin x + cos x )
3                                              2                      3
a)        ∫ cos        xdx ; b)                                       dx ; c)     ∫x         1 − x 2 dx
0                                 0                                    0.5

4. Compute the integrals:
π /2                                    π
4     2                                6
a)           ∫ sin x cos dx ; b)                 ∫ sin       xdx .
0                                   0
π
5. Let I n = ∫ x n cos xdx for all n > 1. Establish the following recurrence relation:
0
π
n −1
I n = −nπ                 − n( n − 1) I n − 2 . Use this to evaluate I 4 = ∫ x 4 cos xdx .
0
∞
n −x
6. If J n =            ∫x       e     dx , for all n > 1 integer, show that J n = nJ n −1 . Deduce that J n = n! .
0

7. Discuss why the following integrals are improper and evaluate those that converge:

Razvan SATNOIANU                                                   Calculus 106              2004
Razvan SATNOIANU                           Calculus 106      2004

1                          ∞
x                      −x
a)   ∫                dx ; b)   ∫e        dx .
0       1 − x2             −∞

8. Find the area of the region bounded by the curve y = x ( x − 1)e − x , the y-axis and the
∞
positive x-axis, i.e. evaluate the integral           ∫ y dx .
0
2      2
 x  y
9. Find the area enclosed by the ellipse   +   = 1 . When a = b this will give you the
a b
area of the circle. Check this particular case to see if you get the answer that you expect.

1. a)                                             (       )
7 + 6 x ; b) − 3 tan x ; c) 9 cos 729 x 3 + sin( x) cos(cos( x 3 )) .

 2                                                            1             1
2.  − , ∞  . Hint: the given integral can be computed directly as ln(2 + 7 x) − ln(2) .
 7                                                            7             7

3. a) 11/24; b) 0.5 + π / 4 ; c) 11 3 / 160 .
π            5π
4. a)          ; b)       .
32           16

5. 4π (6 − π 2 ) .
7. a) integrand infinite when x = 1; the integral is convergent being equal to 1;
b) the integration interval is infinite; the integral is convergent and its value is 2.

8. 6e −1 − 1 .
9. π ab . When a = b then we get the area of the circle π a 2 .

Razvan SATNOIANU                              Calculus 106           2004

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