Selesaikan masing-masing integral parsial berikut ini

a. $\frac{2}{3} x (x-16)^{3/2} - \frac{4}{15} (x-16)^{5/2} + C$, b. $\frac{2}{3} x (x+10)^{3/2} - \frac{4}{15} (x+10)^{5/2} + C$

Pembahasan

Kita akan menyelesaikan integral parsial menggunakan formula $\int u \, dv = uv - \int v \, du$.

a. $\int x \sqrt{x-16} \, dx$
Pilih $u = x$, maka $du = dx$.
Pilih $dv = \sqrt{x-16} \, dx = (x-16)^{1/2} \, dx$. Untuk mencari $v$, kita integralkan $dv$: $v = \int (x-16)^{1/2} \, dx$. Gunakan substitusi $w = x-16$, maka $dw = dx$. Integral menjadi $\int w^{1/2} \, dw = \frac{w^{3/2}}{3/2} = \frac{2}{3} w^{3/2} = \frac{2}{3} (x-16)^{3/2}$.

Menggunakan formula integral parsial:
$\int x \sqrt{x-16} \, dx = x \left( \frac{2}{3} (x-16)^{3/2} \right) - \int \frac{2}{3} (x-16)^{3/2} \, dx$
$= \frac{2}{3} x (x-16)^{3/2} - \frac{2}{3} \int (x-16)^{3/2} \, dx$
Untuk integral $\int (x-16)^{3/2} \, dx$, gunakan substitusi yang sama ($w=x-16$, $dw=dx$): $\int w^{3/2} \, dw = \frac{w^{5/2}}{5/2} = \frac{2}{5} w^{5/2} = \frac{2}{5} (x-16)^{5/2}$.

Maka, hasil integralnya adalah:
$= \frac{2}{3} x (x-16)^{3/2} - \frac{2}{3} \left( \frac{2}{5} (x-16)^{5/2} \right) + C$
$= \frac{2}{3} x (x-16)^{3/2} - \frac{4}{15} (x-16)^{5/2} + C$

b. $\int x \sqrt{x+10} \, dx$
Pilih $u = x$, maka $du = dx$.
Pilih $dv = \sqrt{x+10} \, dx = (x+10)^{1/2} \, dx$. Untuk mencari $v$, kita integralkan $dv$: $v = \int (x+10)^{1/2} \, dx$. Gunakan substitusi $w = x+10$, maka $dw = dx$. Integral menjadi $\int w^{1/2} \, dw = \frac{w^{3/2}}{3/2} = \frac{2}{3} w^{3/2} = \frac{2}{3} (x+10)^{3/2}$.

Menggunakan formula integral parsial:
$\int x \sqrt{x+10} \, dx = x \left( \frac{2}{3} (x+10)^{3/2} \right) - \int \frac{2}{3} (x+10)^{3/2} \, dx$
$= \frac{2}{3} x (x+10)^{3/2} - \frac{2}{3} \int (x+10)^{3/2} \, dx$
Untuk integral $\int (x+10)^{3/2} \, dx$, gunakan substitusi yang sama ($w=x+10$, $dw=dx$): $\int w^{3/2} \, dw = \frac{w^{5/2}}{5/2} = \frac{2}{5} w^{5/2} = \frac{2}{5} (x+10)^{5/2}$.

Maka, hasil integralnya adalah:
$= \frac{2}{3} x (x+10)^{3/2} - \frac{2}{3} \left( \frac{2}{5} (x+10)^{5/2} \right) + C$
$= \frac{2}{3} x (x+10)^{3/2} - \frac{4}{15} (x+10)^{5/2} + C$