1-2-3-x-2-3-3-

Question Number 152887 by otchereabdullai@gmail.com last updated on 02/Sep/21

\int_{1}^{2} \frac{3}{\sqrt{{(x^{2} + 3)}^{3}}}

Answered by Ar Brandon last updated on 02/Sep/21

I = \int_{1}^{2} \frac{3}{\sqrt{{(x^{2} + 3)}^{3}}} d x

= 3 \int_{1}^{2} \frac{d x}{x^{3} \sqrt{{(1 + \frac{3}{x^{2}})}^{3}}}, u = 1 + \frac{3}{x^{2}} \Rightarrow d u = - \frac{6}{x^{3}} d x

= - \frac{1}{2} \int_{4}^{\frac{7}{4}} \frac{d u}{\sqrt{u^{3}}} = - \frac{1}{2} \int_{4}^{\frac{7}{4}} u^{- \frac{3}{2}} d u = {[u^{- \frac{1}{2}}]}_{4}^{\frac{7}{4}}

= (\sqrt{\frac{4}{7}} - 2) = 2 (\frac{1}{\sqrt{7}} - 1)