Tentukan antiturunan untuk setiap fungsi berikut ini. a.

a. 12/5 x^(5/3) - 9 x^(2/3) + C, b. 3/13 x^(13/3) + 9/10 x^(10/3) + 9/7 x^(7/3) + 3/4 x^(4/3) + C.

Pembahasan

a. $\int (4x^{2/3} - 6x^{-1/3}) dx = 4 \int x^{2/3} dx - 6 \int x^{-1/3} dx = 4 \frac{x^{2/3+1}}{2/3+1} - 6 \frac{x^{-1/3+1}}{-1/3+1} + C = 4 \frac{x^{5/3}}{5/3} - 6 \frac{x^{2/3}}{2/3} + C = \frac{12}{5} x^{5/3} - 9 x^{2/3} + C$. 
b. $\int x^{1/3}(x+1)^3 dx = \int x^{1/3}(x^3+3x^2+3x+1) dx = \int (x^{10/3}+3x^{7/3}+3x^{4/3}+x^{1/3}) dx = \frac{x^{13/3}}{13/3} + 3\frac{x^{10/3}}{10/3} + 3\frac{x^{7/3}}{7/3} + \frac{x^{4/3}}{4/3} + C = \frac{3}{13} x^{13/3} + \frac{9}{10} x^{10/3} + \frac{9}{7} x^{7/3} + \frac{3}{4} x^{4/3} + C$.