(x3 - 4x)dx

Quiz 1 (9th September)
(1)
R2
(x3 − 4x)dx =?
−1
Solution:
Z
2
3
(x − 4x)dx
=
−1
2
x4
2
− 2x 4
−1
24
1
− 2.22 − + 2
4
4
1
= 4−8− +2
4
1
= −4 − + 2
4
1
= −2 −
4
9
= − .
4
=
R x2
(2) If g(x) = 1 1t dt, then g 0 (x) =?
Solution: Let F 0 (t) = 1t . Then,
g(x)
= F (x2 ) − F (1)
g 0 (x)
(3)
R √5
= F 0 (x2 ).2x − (F (1))0
1
.2x − 0
g 0 (x) =
x2
2
=
.
x
√
x3 25 − x4 dx =?
Solution: Let u = 25 − x4 . Then, du = −4x3 dx. Thus,
Z
Z p
p
1
3
4
25 − x4 (−4x3 )dx
x 25 − x dx =
−4
Z
1
= −
u1/2 du
4
12
= − u3/2
43
1
= − (25 − x4 )3/2 .
6
Putting the limits in the integration back,
Z √5 p
√
1
1
x3 25 − x4 dx = − (25 − ( 5)4 )3/2 + (25 − 04 )3/2
6
6
0
1
= 0 + 253/2
6
125
=
.
6
0
1

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