dx-4x-2-1-3-

Question Number 213376 by RoseAli last updated on 03/Nov/24

\int \frac{d x}{\sqrt{{(4 x^{2} + 1)}^{3}}}

Commented by Frix last updated on 03/Nov/24

Sometimes just use your brain & experience

\frac{d}{d x} [\frac{g (x)}{\sqrt{4 x^{2} + 1}}] = \frac{g^{'} (x) (4 x^{2} + 1) - 4 g (x)}{{(4 x^{2} + 1)}^{\frac{3}{2}}}

g^{'} (x) (4 x^{2} + 1) - 4 g (x) = 1 \Rightarrow g (x) = x

\int \frac{d x}{\sqrt{{(4 x^{2} + 1)}^{3}}} = \frac{x}{\sqrt{4 x^{2} + 1}} + C

Answered by Frix last updated on 03/Nov/24

\int \frac{d x}{{(4 x^{2} + 1)}^{\frac{3}{2}}} \overset{[t = \tan^{- 1} 2 x]}{=} \frac{1}{2} \int \cos t d t = \frac{1}{2} \sin t =

= \frac{x}{\sqrt{4 x^{2} + 1}} + C

Answered by Frix last updated on 03/Nov/24

\int \frac{d x}{{(4 x^{2} + 1)}^{\frac{3}{2}}} \overset{[t = 2 x + \sqrt{4 x^{2} + 1}]}{=} \int \frac{2 t}{{(t^{2} + 1)}^{2}} d t =

= - \frac{1}{t^{2} + 1} = \frac{x}{\sqrt{4 x^{2} + 1}} + C

Facebook Tweet Pin