find-dt-t-1-t-t-t-1-2-calculate-1-3-dt-t-1-t-t-t-1-

Question Number 44306 by abdo.msup.com last updated on 26/Sep/18

f i n d \int \frac{d t}{(t + 1) \sqrt{t} + t \sqrt{t + 1}}

2) c a l c u l a t e \int_{1}^{3} \frac{d t}{(t + 1) \sqrt{t} + t \sqrt{t + 1}}

Commented by maxmathsup by imad last updated on 29/Sep/18

I = \int \frac{d t}{\sqrt{t} \sqrt{t + 1} (\sqrt{t + 1} + \sqrt{t})} = \int \frac{\sqrt{t + 1} - \sqrt{t}}{\sqrt{t} \sqrt{t + 1}} d t

= \int \frac{d t}{\sqrt{t}} - \int \frac{d t}{\sqrt{t + 1}} = 2 \sqrt{t} - 2 \sqrt{t + 1} + c .

Commented by maxmathsup by imad last updated on 29/Sep/18

\int_{1}^{3} \frac{d t}{(t + 1) \sqrt{t} + t \sqrt{t + 1}} = {[2 \sqrt{t} - 2 \sqrt{t + 1}]}_{1}^{3} = 2 \sqrt{3} - 4 - 2 + 2 \sqrt{2} = 2 \sqrt{2} + 2 \sqrt{3} - 6 .