Question-158674

Question Number 158674 by cortano last updated on 07/Nov/21

Commented by tounghoungko last updated on 07/Nov/21

I_{1} = \int \frac{d x}{\sqrt[3]{x} + \sqrt[4]{x}}; x = r^{12}

I_{1} = \int \frac{12 r^{11}}{r^{4} + r^{3}} d r = \int \frac{12 r^{8}}{r + 1} d r

Commented by tounghoungko last updated on 07/Nov/21

I_{2} = \int \frac{\ln (1 + \sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} d x

l e t x = h^{6} \Rightarrow d x = 6 h^{5} d h

I_{2} = \int \frac{\ln (1 + h)}{h^{2} + h^{3}} (6 h^{5}) d h

I_{2} = \int \frac{\ln (1 + h)}{1 + h} (6 h^{3} d h)

l e t u = \ln (1 + h) \Rightarrow d u = \frac{d h}{1 + h}

\Rightarrow h = e^{u} - 1

I_{2} = 6 \int {(e^{u} - 1)}^{3} u d u

I_{2} = 6 \int u (e^{3 u} - 3 e^{2 u} + 3 e^{u} - 1) d u