Hitunglah integral berikut.integral x^3 akar(x-2) dx

\( \frac{2}{9}(x-2)^{9/2} + \frac{12}{7}(x-2)^{7/2} + \frac{24}{5}(x-2)^{5/2} + \frac{16}{3}(x-2)^{3/2} \) + C

Pembahasan

Untuk menghitung integral \( \int x^3 \sqrt{x-2} dx \), kita dapat menggunakan metode substitusi. Misalkan \(u = x-2\), maka \(x = u+2\) dan \(du = dx\). 
Integral menjadi: \( \int (u+2)^3 \sqrt{u} du \) 
= \( \int (u^3 + 6u^2 + 12u + 8) u^{1/2} du \) 
= \( \int (u^{7/2} + 6u^{5/2} + 12u^{3/2} + 8u^{1/2}) du \) 
Sekarang, kita integralkan suku demi suku:
= \( \frac{u^{9/2}}{9/2} + 6\frac{u^{7/2}}{7/2} + 12\frac{u^{5/2}}{5/2} + 8\frac{u^{3/2}}{3/2} \) + C
= \( \frac{2}{9}u^{9/2} + \frac{12}{7}u^{7/2} + \frac{24}{5}u^{5/2} + \frac{16}{3}u^{3/2} \) + C
Substitusikan kembali \(u = x-2\):
= \( \frac{2}{9}(x-2)^{9/2} + \frac{12}{7}(x-2)^{7/2} + \frac{24}{5}(x-2)^{5/2} + \frac{16}{3}(x-2)^{3/2} \) + C