find f(a) =∫ (x−a)(√(x^2 +a^2 ))dx


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Question Number 62208 by maxmathsup by imad last updated on 17/Jun/19

$f i n d f (a) = \int (x - a) \sqrt{x^{2} + a^{2}} d x$
Commented by maxmathsup by imad last updated on 18/Jun/19
$f(a) =∫ (x−a)(√(x^2 +a^2 ))dx ⇒f(a)=∫ x(√(x^2 +a^2 ))dx−a ∫(√(x^2 +a^2 ))dx ∫ x(√(x^2 +a^2 ))dx = (1/3)(x^2 +a^2 )^(3/2) +c_1 ∫ (√(x^2 +a^2 ))dx =_(x =ash(t)) ∫ ∣a∣ch(t)ach(t)dt =a∣a∣∫ ch^2 t dt =a∣a∣ ∫ ((1+ch(2t))/2)dt =((a∣a∣)/2)t +((a∣a∣)/4) sh(2t) +c_2 =((a∣a∣)/2)t +((a∣a∣)/2)sh(t)ch(t) +c_2 =((a∣a∣)/2){ argsh((x/a)) +(x/a)(√(1+(x^2 /a^2 )))} +c_2 =((a∣a∣)/2){ln((x/a)+(√(1+(x^2 /a^2 )))) +(x/(a∣a∣))(√(a^2 +x^2 ))} +c_2 =((a∣a∣)/2) ln((x/a) +((√(a^2 +x^2 ))/(∣a∣))) +(x/2) (√(a^2 +x^2 )) +c_2 ⇒ f(a) =(1/3)(x^2 +a^2 )^(3/2) −((a^2 ∣a∣)/2)ln((x/a) +((√(a^2 +x^2 ))/(∣a∣)))−((ax)/2)(√(a^2 +x^2 )) +C$
$f (a) = \int (x - a) \sqrt{x^{2} + a^{2}} d x \Rightarrow f (a) = \int x \sqrt{x^{2} + a^{2}} d x - a \int \sqrt{x^{2} + a^{2}} d x$ $\int x \sqrt{x^{2} + a^{2}} d x = \frac{1}{3} {(x^{2} + a^{2})}^{\frac{3}{2}} + c_{1}$ $\int \sqrt{x^{2} + a^{2}} d x =_{x = a s h (t)} \int ∣ a ∣ c h (t) a c h (t) d t = a ∣ a ∣ \int {c h}^{2} t d t$ $= a ∣ a ∣ \int \frac{1 + c h (2 t)}{2} d t = \frac{a ∣ a ∣}{2} t + \frac{a ∣ a ∣}{4} s h (2 t) + c_{2}$ $= \frac{a ∣ a ∣}{2} t + \frac{a ∣ a ∣}{2} s h (t) c h (t) + c_{2} = \frac{a ∣ a ∣}{2} {a r g s h (\frac{x}{a}) + \frac{x}{a} \sqrt{1 + \frac{x^{2}}{a^{2}}}} + c_{2}$ $= \frac{a ∣ a ∣}{2} {l n (\frac{x}{a} + \sqrt{1 + \frac{x^{2}}{a^{2}}}) + \frac{x}{a ∣ a ∣} \sqrt{a^{2} + x^{2}}} + c_{2}$ $= \frac{a ∣ a ∣}{2} l n (\frac{x}{a} + \frac{\sqrt{a^{2} + x^{2}}}{∣ a ∣}) + \frac{x}{2} \sqrt{a^{2} + x^{2}} + c_{2} \Rightarrow$ $f (a) = \frac{1}{3} {(x^{2} + a^{2})}^{\frac{3}{2}} - \frac{a^{2} ∣ a ∣}{2} l n (\frac{x}{a} + \frac{\sqrt{a^{2} + x^{2}}}{∣ a ∣}) - \frac{a x}{2} \sqrt{a^{2} + x^{2}} + C$
Commented by maxmathsup by imad last updated on 18/Jun/19

$i f a = 0 w e g e t f (a) = \int x ∣ x ∣ d x = . . . .$
	Answered by tanmay last updated on 17/Jun/19

	$\int x \sqrt{x^{2} + a^{2}} d x - a \int \sqrt{x^{2} + a^{2}} d x$ $\frac{1}{2} \int {(x^{2} + a^{2})}^{\frac{1}{2}} \times d (x^{2} + a^{2}) - a [\frac{x \sqrt{x^{2} + a^{2}}}{2} + \frac{a^{2}}{2} l n (x + \sqrt{x^{2} + a^{2}})] + c$ $\frac{1}{2} \times \frac{{(x^{2} + a^{2})}^{\frac{3}{2}}}{\frac{3}{2}} - a [\frac{x \sqrt{x^{2} + a^{2}}}{2} + \frac{a^{2}}{2} l n (x + \sqrt{x^{2} + a^{2}}) + c$
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