∫(√(((1+x)/(1−x)) ))(1+x) dx


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Question Number 62907 by aliesam last updated on 26/Jun/19

$\int \sqrt{\frac{1 + x}{1 - x}} (1 + x) d x$
Commented by mathmax by abdo last updated on 26/Jun/19
$let A =∫(√((1+x)/(1−x)))(1+x)dx changement x =cos(2θ) give 2θ =arcosx ⇒ A =∫ (√((2cos^2 (θ))/(2sin^2 θ)))(1+cos(2θ)(−2sin(2θ)dθ =−2∫ ((cosθ)/(sinθ))(1+cos(2θ) 2sinθ cosθ dθ =−4 ∫ cos^2 θ{1+cos(2θ}dθ =−4 ∫ (((1+cos(2θ)^2 ))/2)dθ =−2 ∫ (1+2cos(2θ) +cos^2 (2θ))dθ =−2θ −(4/2) sin(2θ) −2 ∫((1+cos(4θ))/2)dθ =−2θ −2 sin(2θ) −θ −(1/4)sin(4θ) +c =−3θ −2sin(2θ)−(1/2)sin(2θ)cos(2θ)+c =−(3/2) arcosx−2(√(1−x^2 )) −(x/2)(√(1−x^2 )) +c .$
$l e t A = \int \sqrt{\frac{1 + x}{1 - x}} (1 + x) d x c h a n g e m e n t x = c o s (2 θ) g i v e 2 θ = a r c o s x \Rightarrow$ $A = \int \sqrt{\frac{2 {c o s}^{2} (θ)}{2 {s i n}^{2} θ}} (1 + c o s (2 θ) (- 2 s i n (2 θ) d θ$ $= - 2 \int \frac{c o s θ}{s i n θ} (1 + c o s (2 θ) 2 s i n θ c o s θ d θ = - 4 \int {c o s}^{2} θ {1 + c o s (2 θ} d θ$ $= - 4 \int \frac{(1 + c o s {(2 θ)}^{2})}{2} d θ = - 2 \int (1 + 2 c o s (2 θ) + {c o s}^{2} (2 θ)) d θ$ $= - 2 θ - \frac{4}{2} s i n (2 θ) - 2 \int \frac{1 + c o s (4 θ)}{2} d θ$ $= - 2 θ - 2 s i n (2 θ) - θ - \frac{1}{4} s i n (4 θ) + c$ $= - 3 θ - 2 s i n (2 θ) - \frac{1}{2} s i n (2 θ) c o s (2 θ) + c$ $= - \frac{3}{2} a r c o s x - 2 \sqrt{1 - x^{2}} - \frac{x}{2} \sqrt{1 - x^{2}} + c .$
	Answered by Hope last updated on 26/Jun/19

	$\int \frac{{(1 + x)}^{2}}{\sqrt{1 - x^{2}}} d x$ $\int \frac{d x}{\sqrt{1 - x^{2}}} + \int \frac{2 x d x}{\sqrt{1 - x^{2}}} + \int \frac{x^{2}}{\sqrt{1 - x^{2}}} d x$ $x = s i n a d x = c o s a d a$ $\int \frac{c o s a d a}{c o s a} + \int \frac{2 s i n a \times c o s a d a}{c o s a} + \int \frac{{s i n}^{2} a \times c o s a d a}{c o s a}$ $\int d a + 2 \int s i n a d a + \int \frac{1 - c o s 2 a}{2} d a$ $\int d a + 2 \int s i n a d a + \frac{1}{2} \int d a - \frac{1}{2} \int c o s 2 a d a$ $a - 2 c o s a + \frac{1}{2} a - \frac{1}{2} \times \frac{s i n 2 a}{2} + c$ $= \frac{3}{2} {s i n}^{- 1} (x) - 2 \sqrt{1 - x^{2}} - \frac{1}{4} \times 2 x \times \sqrt{1 - x^{2}} + c$
	Commented by mathmax by abdo last updated on 26/Jun/19

	$t h e a n s w e r o f s i r h o p e i s c o r r e c t .$
	Commented by aliesam last updated on 26/Jun/19

	$\sqrt{1 + x} (1 + x) \neq {(1 + x)}^{2}$
	Commented by peter frank last updated on 26/Jun/19

	$\sqrt{\frac{1 + x}{1 - x} \times \frac{1 + x}{1 + x}} (1 + x)$ $\frac{1 + x}{\sqrt{1 - x^{2}}} . (1 + x)$ $\frac{{(1 + x)}^{2}}{\sqrt{1 - x^{2}}}$
	Commented by aliesam last updated on 27/Jun/19

	$o h t h a t s r i g h t s o s o r r y s i r$
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