Ingat Aturan Rantai pada Turunan :

Jika kedua ruas diintegralkan, maka diperoleh

)('))(('))(( xgxgfxgfdx

d

dxxgxgfdxxgfdx

d)('))(('))((

dxxgxgfCxgf )('))(('))((

dari definisi integral tak tentu

Misal u = g(x), maka du = g’(x)dx

Disubstitusi ke atas diperoleh

Cxgfdxxgxgf ))(()('))(('

Cufduuf )()('

1. Mulai dengan fungsi yang diintegralkan2. Kita misalkan u = g(x)3. Hitung du4. Substitusi u dan du5. Integralkan6. Ganti u dengan g(x)

Hitunglah

JawabMisalkan u = 3x + 5 , maka du = 3 dx , dx

= 1/3 duSubstitusi ke fungsi di atas diperoleh

dxx )53sin(

CxCuududxx )53cos(cossin)53sin(

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

9dxxe x 53 2

9

Hitunglah

Jawab

Misalkan u = cos x , maka du = -sin x dx atau

sin x dx = -du.Sehingga

xdxtan

dxx

xxdx

cos

sintan

CxCxCuu

dudxx

xxdx

seclncoslnln

cos

sintan

Exercise

Bentuk integral dapat

diselesaikan dengan metode Integral By Parts (Integral sebagian – sebagian) , yaitu

dxxfxgxgxfdxxgxf )(')()()()(')(

dxxgxf )()(

Atau lebih dikenal dengan rumus

duvuvdvu

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( 41

41 dxxxxdxxx

Hitunglah dxxx )ln()5( 3

dxxe x )cos(2

dxxx )4cos(2

a

b

c

Exercise

Link to James Stewart

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.

dxxQ

xP )(

)(

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.

kk

rx

A

rx

A

rx

A

)()()( 221

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)

kkk

cbxax

CxB

cbxax

CxB

cbxax

CxB

)()( 22222

211

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)

9. Integrate linear factors:

rxAdx

rx

A

ln)( 1

1

111 )(1)(

n

nrx

n

Adx

rx

Afor n > 1

10. Integrate quadratic factors: Some simple formulas:

a

x

A

Cax

Bdx

ax

CBxarctan)ln(

222

22

a

x

a

C

axa

BaCxdx

ax

CBxarctan

2)(2)( 3222

2

222

Hitunglah

Jawab Link Ex1-Int.Fractional

dxxx

xx

214

162

23

Exercise Link to Drii – Int.Fractional

Link to Strategi Pengintegralan

Evaluate

Answer

Evaluate

Answer

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Answer

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Answer

Evaluate

Answer

Evaluate

Answer

Evaluate

Answer

Evaluate

Answer

Evaluate

Answer

Link to Tabel Rumus Umum integral