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Question Number 93638 by i jagooll last updated on 14/May/20

\int \frac{x^{4} + 4 x^{2}}{\sqrt{x^{2} + 4}} d x ?

Commented by i jagooll last updated on 14/May/20

thank you both

Commented by mathmax by abdo last updated on 14/May/20

I = \int \frac{x^{4} + 4 x^{2}}{\sqrt{x^{2} + 4}} d x c h a n g e m e n t \sqrt{x^{2} + 4} = t g i v e x^{2} = t^{2} - 4 \Rightarrow x d x = t d t

I = \int \frac{{(t^{2} - 4)}^{3} + 4 \sqrt{t^{2} - 4}}{t} t d t = \int ({(t^{2} - 4)}^{3} + 4 \sqrt{t^{2} - 4}) d t

= \int {(t^{2} - 4)}^{3} d t + 4 \int \sqrt{t^{2} - 4} d t b u t

\int {(t^{2} - 4)}^{3} d t = \int (t^{6} - 3 t^{4} (4) + 3 t^{2} {(4)}^{2} - 4^{3}) d t

= \frac{t^{7}}{7} - \frac{12}{5} t^{5} + \frac{48}{3} t^{3} - 64 t + c_{1}

\int \sqrt{t^{2} - 4} d t =_{t = 2 c h z} \int 2 s h z (2 s h z) d z = 4 \int {s h}^{2} z d z

= 2 \int (c h (2 z) - 1) d z = s h (2 z) - 2 z + c_{2}

= 2 s h (z) c h (z) - 2 z + c_{2}

= 2 \sqrt{{c h}^{2} z - 1} \times \frac{t}{2} - 2 a r g c h (\frac{t}{2}) + c_{2}

= t \sqrt{\frac{t^{2}}{4} - 1} - 2 l n (\frac{t}{2} + \sqrt{\frac{t^{2}}{4} - 1}) + c_{2} \Rightarrow

I = \frac{1}{7} {(\sqrt{x^{2} + 4})}^{7} - \frac{12}{5} {(\sqrt{x^{2} + 4})}^{5} + 16 {(\sqrt{x^{2} + 4})}^{3} - 64 \sqrt{x^{2} + 4}

+ 2 t \sqrt{t^{2} - 4} - 2 l n (t + \sqrt{t^{2} - 4}) + C

Answered by Ar Brandon last updated on 14/May/20

L = \int \frac{x^{4} + 4 x^{2}}{\sqrt{x^{2} + 4}} d x = \int x^{2} \cdot \frac{x^{2} + 4}{\sqrt{x^{2} + 4}} d x = \int x^{2} \sqrt{x^{2} + 4} d x

x = 2 \sinh θ \Rightarrow d x = 2 \cosh θ d θ

L = \int {4 \sinh}^{2} θ \cdot \sqrt{{4 \sinh}^{2} θ + 4} \cdot 2 \cosh θ d θ

\Rightarrow L = 4 \int {4 \sinh}^{2} θ \cosh^{2} θ d θ = 4 \int \sinh^{2} 2 θ d θ

\Rightarrow L = 2 \int (\cosh 4 θ - 1) = \frac{\sinh 4 θ}{2} - 2 θ + C

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

Answered by Kunal12588 last updated on 14/May/20

I = \int \frac{x^{2} (x^{2} + 4)}{\sqrt{x^{2} + 4}} d x

x^{2} + 4 = t

\Rightarrow d x = \frac{1}{2 \sqrt{t - 4}} d t

I = \frac{1}{2} \int \frac{(t - 4) t}{\sqrt{t} \sqrt{t - 4}} d t = \frac{1}{2} \int \sqrt{t} \sqrt{t - 4} d t = \frac{1}{2} \int \sqrt{t^{2} - 4 t} d t

= \frac{1}{2} \int \sqrt{{(t - 2)}^{2} - 4} d t

= \frac{1}{2} [\frac{1}{2} (t - 2) \sqrt{t^{2} - 4 t} - \frac{4}{2} l n ∣ t - 2 + \sqrt{t^{2} - 4 t} ∣] + C

= \frac{1}{2} [\frac{1}{2} x (x^{2} + 2) \sqrt{x^{2} + 4} - 2 l n ∣ x^{2} + 2 + x \sqrt{x^{2} + 4} ∣] + C

= \frac{1}{4} x (x^{2} + 2) \sqrt{x^{2} + 4} - l n ∣ x^{2} + x \sqrt{x^{2} + 4} + 2 ∣ + C