dx-x-x-1-3-

Question Number 118347 by bramlexs22 last updated on 17/Oct/20

\int \frac{d x}{\sqrt{x} + \sqrt[3]{x}}

Answered by benjo_mathlover last updated on 17/Oct/20

s e t t i n g x = r^{6}

\int \frac{6 r^{5}}{r^{3} + r^{2}} d r = \int \frac{6 r^{3}}{r + 1} d r = 6 \int (r^{2} - r + 1) - \frac{1}{r + 1} d r

= 6 (\frac{1}{3} r^{3} - \frac{1}{2} r^{2} + r - \ln ∣ r + 1 ∣) + c

= 2 r^{3} - 3 r^{2} + 6 r - 6 \ln ∣ r + 1 ∣ + c

= 2 \sqrt{x} - 3 \sqrt[3]{x} + 6 \sqrt[6]{x} - 6 \ln ∣ \sqrt[6]{x} + 1 ∣ + c