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2.  Ingat Aturan Rantai pada Turunan :
 Jika kedua ruas diintegralkan, maka diperoleh
)('))(('))(( xgxgfxgf
dx
d
=
dxxgxgfdxxgf
dx
d
)('))(('))(( ∫∫ =
dxxgxgfCxgf )('))(('))(( ∫=+
dari definisi integral tak tentu

3.  Misal u = g(x), maka du = g’(x)dx
 Disubstitusi ke atas diperoleh
Cxgfdxxgxgf +=∫ ))(()('))(('
Cufduuf +=∫ )()('

4. 1. Mulai dengan fungsi yang diintegralkan
2. Kita misalkan u = g(x)
3. Hitung du
4. Substitusi u dan du
5. Integralkan
6. Ganti u dengan g(x)

5. Hitunglah
Jawab
Misalkan u = 3x + 5 , maka du = 3 dx , dx = 1/3 du
Substitusi ke fungsi di atas diperoleh
dxx )53sin( +∫
CxCuududxx ++−=+−==+ ∫∫ )53cos(cossin)53sin(

6.  Hitunglah
 Jawab
 Misalkan u = -3x2
+ 5 , maka du = -6x dx atau
x dx = -1/6 du
dxxe x 53 2
9 +−
∫
CeCedue xuu
+−=+−=−= +−
∫
53 2
6
9
6
9
6
9
dxxe x 53 2
9 +−
∫

7.  Hitunglah
 Jawab
 Misalkan u = cos x , maka du = -sin x dx atau
sin x dx = -du.
Sehingga

xdxtan∫
dx
x
x
xdx ∫∫ =
cos
sin
tan
CxCxCu
u
du
dx
x
x
xdx +=+−=+−=
−
== ∫∫∫ seclncoslnln
cos
sin
tan

8.  Exercise

9.  Bentuk integral dapat
 diselesaikan dengan metode Integral By Parts
(Integral sebagian – sebagian) , yaitu
dxxfxgxgxfdxxgxf ∫∫ −= )(')()()()(')(
dxxgxf∫ )()(
Atau lebih dikenal dengan rumus
duvuvdvu ∫∫ −=

10.  Hitunglah
 Jawab
 Misalkan u = 3 – 5x , du = -5 dx.
dv = cos 4x , v = ¼ sin 4x dx
 Maka
dxxx )4cos()53(∫ −
∫∫ −−−=− )5)(4sin()4sin()(53()4cos()53( 4
1
4
1
dxxxxdxxx

11.  Hitunglah dxxx )ln()5( 3
∫ +
dxxe x
)cos(2
∫
dxxx )4cos(2
∫
a
b
c
Exercise

12.  Link to James Stewart

13.  The method of Partial Fractions provides a way
to integrate all rational functions. Recall that a
rational function is a function of the form
where P and Q are polynomials.
1. The technique requires that the degree of the
numerator (pembilang) be less than the degree
of the denominator (penyebut)
If this is not the case then we first must divide
the numerator into the denominator.
dx
xQ
xP
∫ )(
)(

14. 2. We factor the denominator Q into powers of
distinct linear terms and powers of distinct
quadratic polynomials which do not have real
roots.
3. If r is a real root of order k of Q, then the partial
fraction expansion of P/Q contains a term of the
form

where A1, A2, ..., Ak are unknown constants.
k
k
rx
A
rx
A
rx
A
)()()( 2
21
−
++
−
+
−


15. 4. If Q has a quadratic factor ax2
+ bx + c which
corresponds to a complex root of order k, then the
partial fraction expansion of P/Q contains a term of
the form

 where B1, B2, ..., Bk and C1, C2, ..., Ck are
unknown constants.
5. After determining the partial fraction expansion of
P/Q, we set P/Q equal to the sum of the terms of
the partial fraction expansion. (See Ex-2.Int.Frac)
k
kk
cbxax
CxB
cbxax
CxB
cbxax
CxB
)()( 222
22
2
11
++
+
++
++
+
+
++
+


16. 6. We then multiply both sides by Q to get some
expression which is equal to P.
7. Now, we use the property that two polynomials
are equal if and only if the corresponding
coefficients are equal.
(see ex3-int.Fractional)
8. We express the integral of P/Q as the sum of
the integrals of the terms of the partial fraction
expansion.
(see Ex4-Int.Fractional)

17. 9. Integrate linear factors:
rxAdx
rx
A
−=
−∫ ln
)(
1
1
111
)(
1)(
+−
−
+−
=
−∫
n
n
rx
n
A
dx
rx
A
for n > 1

18. 10. Integrate quadratic factors:
Some simple formulas:






++=
+
+
∫ a
x
A
C
ax
B
dx
ax
CBx
arctan)ln(
2
22
22






+
+
−
=
+
+
∫ a
x
a
C
axa
BaCx
dx
ax
CBx
arctan
2)(2)( 3222
2
222

19.  Hitunglah
 Jawab
 Link Ex1-Int.Fractional
dx
xx
xx
214
16
2
23
−+
−+
∫

20.  Exercise
 Link to Drii – Int.Fractional

21.  Link to Strategi Pengintegralan

22.  Evaluate
 Answer

23.  Evaluate
 Answer

24.  Evaluate
 Answer

25.  Evaluate
 Answer

26.  Evaluate
 Answer

27.  Evaluate
 Answer

28.  Evaluate
 Answer

29.  Evaluate
 Answer

30.  Evaluate
 Answer

31.  Link to Tabel Rumus Umum integral