082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

442 Chapter 13 Integration
( )
1 t
(r)
5 tan−1 +c
5
( )
(s) sin −1 t
+c
5
(t) 6tan −1 x + x 6 + x3
18 +c
6 Speed:t + t2 4 +c,distance: t2 2 + t3
12 +ct+d
7 (a) 2t +e −t +c (b) 5.0498
8 (a) coshax +c
a
(b) sinhax +c
a
(c) 3 2 cosh2x + 1 sinh4x +c
4
9 100(10t +e −t ) +c
( )
1 x
10 (a)
2 tan−1 +c
2
( )
1
(b)
2 √ x
2 tan−1 √ +c
2
(c)
3
√
2
tan −1 ( √ 2x)+c
( )
(d) sin −1 x
+c
3
( )
(e) 2sin −1 x
+c
2
(√ )
−7 3
(f) √ sin −1 √ x +c
3 2
(g) √ ( )
2sin −1 x
√ +c
2
11 3ln|x|+x+c
12 (a)
1
x + 2
x 2 +1
(b) ln|x| +2tan −1 x +c
13 (a) 2t −2e −t/2 +c (b) − 1 2 e−t/2
15 (a) −t −2 −ln|t|+c
(b) − 4 3 e−3t −e −t +c
1
(c)
4 ln|sin4x|+c
1
(d) ln|cosec3x −cot3x| +c
6
(e) x+tan −1 x+c
(f) t+c
13.3 DEFINITEANDINDEFINITEINTEGRALS
Alltheintegrationsolutionssofarencounteredhavecontainedaconstantofintegration.
Such integrals are known as indefinite integrals. Integration can be used to determine
the area under curves and this gives rise todefinite integrals.
To estimate the area under y(x), it is divided into thin rectangles. The sum of the
rectangularareasisanapproximationtotheareaunderthecurve.Severalthinrectangles
will give a better approximationthanafewwide ones.
ConsiderFigure13.6wheretheareaisapproximatedbyalargenumberofrectangles.
Suppose each rectangle has width δx. The area of rectangle 1 is y(x 2
)δx, the area of
rectangle2isy(x 3
)δxandsoon.LetA(x n
)denotethetotalareaunderthecurvefromx 1
tox n
. Then,
A(x n
) ≈ sum ofthe rectangularareas =
n∑
y(x i
)δx
i=2
Lettheareabeincreasedbyextendingthebasefromx n
tox.ThenA(x)isthetotalarea
under the curve fromx 1
tox(see Figure 13.7). Then,
increaseinarea = δA =A(x) −A(x n
) ≈y(x)δx