TEKNIK PENGINTEGRALAN ALFITH, S.Pd, M.Pd

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

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

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)

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

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

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

Exercise

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

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

Example 5 Hitunglah a b c Exercise

Reduction Formulas Link to James Stewart

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.

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.

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)

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: for n > 1

10. Integrate quadratic factors: Some simple formulas:

Example 6 Hitunglah Jawab Link Ex1-Int.Fractional

Exercise Link to Drii – Int.Fractional

Strategi Pengintegralan Link to Strategi Pengintegralan

Example 7 Evaluate Answer

Example 8 Evaluate Answer

Example 9 Evaluate Answer

Example 10 Evaluate Answer

Example 11 Evaluate Answer

Example 12 Evaluate Answer

Example 12 Evaluate Answer

Example 13 Evaluate Answer

Example 14 Evaluate Answer

Tabel Rumus Umum Pengintegralan Link to Tabel Rumus Umum integral