Page 302 - Calculus Demystified

Chapter 4
! " 8 289
2
8 sin x 3

3
3
2
(f) x · cos x sin x dx =
4 6
4
2
2
2 3
2 3
sin 8 sin 4 sin 512 sin 64
= − = − .
6 6 6 6
5 x 3 x 2
5
2
5. (a) Area = x + x + 6 dx = + + 6x
2 3 2 2
3 2 3 2
5 5 2 2 405
= + + 6 · 5 − + + 6 · 2 = .
3 2 3 2 6
! " π/4
2
π/4 sin x

(b) Area = sin x cos xdx =
0 2
0
2
2
sin π/4 sin 0 1
= − = .
2 2 4
! " 2
2 2 e x 2 e 2 2 e 1 2 e 4 e
(c) Area = xe x dx = = − = − .
1 2 2 2 2 2
1
! " e
2
2
2

e ln x ln x ln e ln 1 1 0 1
(d) Area = dx = = − = − = .
1 x 2 2 2 2 2 2
1
6.

π π
ε
π
π
ε
π
π
Fig. S4.6(b)
Fig. S4.6(a)