TEKNIK PENGINTEGRALAN

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1 TEKNIK PENGINTEGRALAN
ALFITH, S.Pd, M.Pd

2 Integral dengan Substitusi
Ingat Aturan Rantai pada Turunan : Jika kedua ruas diintegralkan, maka diperoleh dari definisi integral tak tentu

3 Selanjutnya….. Misal u = g(x), maka du = g’(x)dx
Disubstitusi ke atas diperoleh

4 Langkah – langkah integral dg substitusi
Mulai dengan fungsi yang diintegralkan Kita misalkan u = g(x) Hitung du Substitusi u dan du Integralkan Ganti u dengan g(x)

5 Example 1 Hitunglah Jawab
Misalkan u = 3x + 5 , maka du = 3 dx , dx = 1/3 du Substitusi ke fungsi di atas diperoleh

6 Example 2 Hitunglah Jawab
Misalkan u = -3x2 + 5 , maka du = -6x dx atau x dx = -1/6 du

7 Example 3 Hitunglah Jawab
Misalkan u = cos x , maka du = -sin x dx atau sin x dx = -du. Sehingga

8 Exercise

9 Integration by Parts Bentuk integral dapat
diselesaikan dengan metode Integral By Parts (Integral sebagian – sebagian) , yaitu Atau lebih dikenal dengan rumus

10 Example 4 Hitunglah Jawab Misalkan u = 3 – 5x , du = -5 dx.
dv = cos 4x , v = ¼ sin 4x dx Maka

11 Example 5 Hitunglah a b c Exercise

12 Reduction Formulas Link to James Stewart

13 Partial Fractions 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. 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.

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.

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)

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:
for n > 1

18 10. Integrate quadratic factors:
Some simple formulas:

19 Example 6 Hitunglah Jawab Link Ex1-Int.Fractional

20 Exercise Link to Drii – Int.Fractional

21 Strategi Pengintegralan
Link to Strategi Pengintegralan

22 Example 7 Evaluate Answer

23 Example 8 Evaluate Answer

24 Example 9 Evaluate Answer

25 Example 10 Evaluate Answer

26 Example 11 Evaluate Answer

27 Example 12 Evaluate Answer

28 Example 12 Evaluate Answer

29 Example 13 Evaluate Answer

30 Example 14 Evaluate Answer

31 Tabel Rumus Umum Pengintegralan
Link to Tabel Rumus Umum integral