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 Ingat Aturan Rantai pada Turunan :  Jika kedua ruas diintegralkan, maka diperoleh dari definisi integral tak tentu.

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Presentasi berjudul: " Ingat Aturan Rantai pada Turunan :  Jika kedua ruas diintegralkan, maka diperoleh dari definisi integral tak tentu."— Transcript presentasi:

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

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

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

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

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

8  Exercise

9  Bentuk integraldapat  diselesaikan dengan metode Integral By Parts (Integral sebagian – sebagian), yaitu Atau lebih dikenal dengan rumus

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

11  Hitunglah 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.

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 A 1, A 2,..., A k are unknown constants.

15 4. If Q has a quadratic factor ax 2 + 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 B 1, B 2,..., B k and C 1, C 2,..., C k 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  Hitunglah  Jawab  Link Ex1-Int.Fractional

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 Link to Tabel Rumus Umum integral


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