Pertanyaan

Selesaikan masing-masing integral parsial berikut ini dengan formula: ∫ u ⋅ dv = u ⋅ v − ∫ v du . a. ∫ x x − 16 dx

Selesaikan masing-masing integral parsial berikut ini dengan formula:

$\int u \cdot dv = u \cdot v - \int v du$ .

a. $\int x x - 16 dx$

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P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Pembahasan

Misalkan: u = x d v = x − 16 d v = ( x − 16 ) 2 1 → d x d u = 1 d u = d x → v = ∫ ( x − 16 ) 2 1 d x = ( 2 1 + 1 ) 1 ⋅ ( x − 16 ) 2 1 + 1 = 2 3 1 ⋅ ( x − 16 ) 2 3 = 3 2 ( x − 16 ) 2 3 Sehingga diperoleh: ∫ x x − 16 dx = = = = = u ⋅ v − ∫ v du x ( 3 2 ( x − 16 ) 2 3 ) − ∫ ( x − 16 ) 2 3 d x 3 2 x ⋅ ( x − 16 ) 2 3 − ( 2 3 + 1 ) 1 ⋅ ( x − 16 ) 2 3 + 1 + C 3 2 x ⋅ ( x − 16 ) 2 3 − 5 2 ⋅ ( x − 16 ) 2 5 + C 3 2 x ⋅ ( x − 16 ) ( x − 16 ) − 5 2 ( x − 16 ) 2 ( x − 16 ) + C Dengan demikian, hasil dari:

Misalkan:

$u = x d v = x - 16 d v = (x - 16)^{\frac{1}{2}} \to \frac{d u}{d x} = 1 d u = d x \to v = \int (x - 16)^{\frac{1}{2}} d x = \frac{1}{( \frac{1}{2} + 1 )} \cdot (x - 16)^{\frac{1}{2} + 1} = \frac{1}{\frac{3}{2}} \cdot (x - 16)^{\frac{3}{2}} = \frac{2}{3} (x - 16)^{\frac{3}{2}}$

Sehingga diperoleh:

$\int x x - 16 dx = = = = = u \cdot v - \int v du x (\frac{2}{3} (x - 16)^{\frac{3}{2}}) - \int (x - 16)^{\frac{3}{2}} d x \frac{2}{3} x \cdot (x - 16)^{\frac{3}{2}} - \frac{1}{( \frac{3}{2} + 1 )} \cdot (x - 16)^{\frac{3}{2} + 1} + C \frac{2}{3} x \cdot (x - 16)^{\frac{3}{2}} - \frac{2}{5} \cdot (x - 16)^{\frac{5}{2}} + C \frac{2}{3} x \cdot (x - 16) (x - 16) - \frac{2}{5} (x - 16)^{2} (x - 16) + C$