1)calculate f(x)=∫_0 ^1 t^2 (√(x^2 +t^2 ))dt with x>0 2) calculste g(x)=∫


Question and Answers Forum

All Questions Topic List
Integration Questions
Previous in All Question Next in All Question
Previous in Integration Next in Integration
Question Number 74637 by mathmax by abdo last updated on 28/Nov/19

$1) c a l c u l a t e f (x) = \int_{0}^{1} t^{2} \sqrt{x^{2} + t^{2}} d t w i t h x > 0$ $2) c a l c u l s t e g (x) = \int_{0}^{1} \frac{t^{2}}{\sqrt{x^{2} + t^{2}}} d t$
Commented by mathmax by abdo last updated on 28/Nov/19
$1) we have f(x)=∫_0 ^1 t^2 (√(x^2 +t^2 ))dt ⇒f(x)=_(t=x sh(u)) ∫_0 ^(argsh((1/x))) x^2 sh^2 u(xch(u))xch(u)du = x^4 ∫_0 ^(ln((1/x)+(√(1+(1/x^2 ))))) sh^2 (u)ch^2 (u)du =(x^4 /4) ∫_0 ^(ln(((1+(√(x^2 +1)))/x))) sh^2 (2u) du =(x^4 /8) ∫_0 ^(ln(((1+(√(x^2 +1)))/x))) (ch(4u)−1))du =(x^4 /(32))[ sh(4u)]_0 ^(ln(((1+(√(x^2 +1)))/x))) −(x^4 /8)ln(((1+(√(x^2 +1)))/x)) =(x^4 /(64))[e^(4u) −e^(−4u) ]_0 ^(ln(((1+(√(x^2 +1)))/x))) −(x^4 /8)ln(((1+(√(x^2 +1)))/x)) =(x^4 /(64)){(((1+(√(x^2 +1)))/x))^4 −(((1+(√(x^2 +1)))/x))^(−4) }−(x^4 /8)ln(((1+(√(x^2 +1)))/x))$
$1) w e h a v e f (x) = \int_{0}^{1} t^{2} \sqrt{x^{2} + t^{2}} d t \Rightarrow f (x) =_{t = x s h (u)} \int_{0}^{a r g s h (\frac{1}{x})} x^{2} {s h}^{2} u (x c h (u)) x c h (u) d u$ $= x^{4} \int_{0}^{l n (\frac{1}{x} + \sqrt{1 + \frac{1}{x^{2}}})} {s h}^{2} (u) {c h}^{2} (u) d u$ $= \frac{x^{4}}{4} \int_{0}^{l n (\frac{1 + \sqrt{x^{2} + 1}}{x})} {s h}^{2} (2 u) d u = \frac{x^{4}}{8} \int_{0}^{l n (\frac{1 + \sqrt{x^{2} + 1}}{x})} (c h (4 u) - 1)) d u$ $= \frac{x^{4}}{32} {[s h (4 u)]}_{0}^{l n (\frac{1 + \sqrt{x^{2} + 1}}{x})} - \frac{x^{4}}{8} l n (\frac{1 + \sqrt{x^{2} + 1}}{x})$ $= \frac{x^{4}}{64} {[e^{4 u} - e^{- 4 u}]}_{0}^{l n (\frac{1 + \sqrt{x^{2} + 1}}{x})} - \frac{x^{4}}{8} l n (\frac{1 + \sqrt{x^{2} + 1}}{x})$ $= \frac{x^{4}}{64} {{(\frac{1 + \sqrt{x^{2} + 1}}{x})}^{4} - {(\frac{1 + \sqrt{x^{2} + 1}}{x})}^{- 4}} - \frac{x^{4}}{8} l n (\frac{1 + \sqrt{x^{2} + 1}}{x})$
	Answered by MJS last updated on 28/Nov/19

	$both with the same substitution$ $t = x \sinh \ln u = \frac{x (u^{2} - 1)}{2 u} \to d t = \frac{x (u^{2} + 1)}{2 u^{2}} d u$ $\int t^{2} \sqrt{x^{2} + t^{2}} d t = \frac{x^{4}}{16} \int \frac{{(u^{4} - 1)}^{2}}{u^{5}} d u =$ $= x^{4} (\frac{u^{8} - 1}{64 u^{4}} - \frac{1}{8} \ln u) = f (x)$ $\int \frac{t^{2}}{\sqrt{x^{2} + t^{2}}} d t = \frac{x^{2}}{4} \int \frac{{(u^{2} - 1)}^{2}}{u^{3}} d u =$ $= x^{2} (\frac{u^{4} - 1}{8 u^{2}} - \frac{1}{2} \ln u) = g (x)$ $the borders : 0 ⩽ t ⩽ 1 \Rightarrow 1 ⩽ u ⩽ \frac{1 + \sqrt{x^{2} + 1}}{x}$ $f (1) = g (1) = 0$ $f (x) = x^{4} (\frac{u^{8} - 1}{64 u^{4}} - \frac{1}{8} \ln u) with u = \frac{1 + \sqrt{x^{2} + 1}}{x}$ $g (x) = x^{2} (\frac{u^{4} - 1}{8 u^{2}} - \frac{1}{2} \ln u) with u = \frac{1 + \sqrt{x^{2} + 1}}{x}$ $sorry I^{'} ve got no time to insert these and$ $transform$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum