calculate-1-2-dx-x-x-x-

Question Number 32302 by abdo imad last updated on 22/Mar/18

c a l c u l a t e \int_{1}^{2} \frac{d x}{x + x \sqrt{x}} .

Answered by sma3l2996 last updated on 23/Mar/18

l e t t = \sqrt{x} \Rightarrow d x = 2 t d t

A = \int_{1}^{2} \frac{d x}{x + x \sqrt{x}} = 2 \int_{1}^{\sqrt{2}} \frac{t d t}{t^{2} (1 + t)} = 2 \int_{1}^{\sqrt{2}} \frac{d t}{t (1 + t)} = 2 \int_{1}^{\sqrt{2}} (\frac{1}{t} - \frac{1}{t + 1}) d t

= 2 {[l n ∣ \frac{t}{t + 1} ∣]}_{1}^{\sqrt{2}} = 2 (l n (\frac{\sqrt{2}}{\sqrt{2} + 1}) + l n 2)