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2t-1-t-4-1-t-dt-

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Question Number 94735 by student work last updated on 20/May/20
∫((2t)/((1+t^4 )(1+t)))dt=?
$$\int\frac{\mathrm{2t}}{\left(\mathrm{1}+\mathrm{t}^{\mathrm{4}} \right)\left(\mathrm{1}+\mathrm{t}\right)}\mathrm{dt}=? \\ $$
Commented by student work last updated on 20/May/20
who can another?
$$\mathrm{who}\:\mathrm{can}\:\mathrm{another}? \\ $$
Commented by student work last updated on 20/May/20
who can solve describly?
$$\mathrm{who}\:\mathrm{can}\:\mathrm{solve}\:\mathrm{describly}? \\ $$
Commented by MJS last updated on 20/May/20
are you able to decompose? I gave you enough hints. if you don′t know decomposition start with some easier examples. I′m tired of typing obvious things.
$$\mathrm{are}\:\mathrm{you}\:\mathrm{able}\:\mathrm{to}\:\mathrm{decompose}?\:\mathrm{I}\:\mathrm{gave}\:\mathrm{you} \\ $$$$\mathrm{enough}\:\mathrm{hints}.\:\mathrm{if}\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{decomposition} \\ $$$$\mathrm{start}\:\mathrm{with}\:\mathrm{some}\:\mathrm{easier}\:\mathrm{examples}. \\ $$$$\mathrm{I}'\mathrm{m}\:\mathrm{tired}\:\mathrm{of}\:\mathrm{typing}\:\mathrm{obvious}\:\mathrm{things}. \\ $$
Commented by student work last updated on 20/May/20
why you not solve?
$$\mathrm{why}\:\mathrm{you}\:\mathrm{not}\:\mathrm{solve}? \\ $$
Commented by MJS last updated on 20/May/20
I solved it. Question 94574 I decomposed it in 2 steps. what else do you want?
$$\mathrm{I}\:\mathrm{solved}\:\mathrm{it}.\:\mathrm{Question}\:\mathrm{94574} \\ $$$$\mathrm{I}\:\mathrm{decomposed}\:\mathrm{it}\:\mathrm{in}\:\mathrm{2}\:\mathrm{steps}.\:\mathrm{what}\:\mathrm{else}\:\mathrm{do}\:\mathrm{you} \\ $$$$\mathrm{want}? \\ $$
Commented by Tony Lin last updated on 21/May/20
∫((2t)/((1+t^4 )(1+t)))dt =∫ ((t^3 −t^2 +t+1)/(1+t^4 ))dt−∫(1/(1+t))dt =(1/(2(√2)))(∫ ((((√2)+2)t+(√2))/(t^2 +(√2)t+1))dt+∫((((√2)−2)t+(√2))/(t^2 −(√2)t+1))dt)−∫(1/(1+t))dt now it is easy to solve it can be expressed by ln and arctan
$$\int\frac{\mathrm{2}{t}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)\left(\mathrm{1}+{t}\right)}{dt} \\ $$$$=\int\:\frac{{t}^{\mathrm{3}} −{t}^{\mathrm{2}} +{t}+\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}−\int\frac{\mathrm{1}}{\mathrm{1}+{t}}{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\int\:\frac{\left(\sqrt{\mathrm{2}}+\mathrm{2}\right){t}+\sqrt{\mathrm{2}}}{{t}^{\mathrm{2}} +\sqrt{\mathrm{2}}{t}+\mathrm{1}}{dt}+\int\frac{\left(\sqrt{\mathrm{2}}−\mathrm{2}\right){t}+\sqrt{\mathrm{2}}}{{t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}+\mathrm{1}}{dt}\right)−\int\frac{\mathrm{1}}{\mathrm{1}+{t}}{dt} \\ $$$${now}\:{it}\:{is}\:{easy}\:{to}\:{solve} \\ $$$${it}\:{can}\:{be}\:{expressed}\:{by}\:{ln}\:{and}\:{arctan} \\ $$
Commented by student work last updated on 21/May/20
explain me
$$\mathrm{explain}\:\mathrm{me} \\ $$
Commented by MJS last updated on 21/May/20
Sir Tony Lin, this is exactly what I answered to the original question
$$\mathrm{Sir}\:\mathrm{Tony}\:\mathrm{Lin},\:\mathrm{this}\:\mathrm{is}\:\mathrm{exactly}\:\mathrm{what}\:\mathrm{I}\:\mathrm{answered} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{original}\:\mathrm{question} \\ $$
Commented by MJS last updated on 21/May/20
Sir student work, if you cannot solve it from there, better start learning basic integrals. there′s no use solving problems like this one for you if you cannot understand each simple step.
$$\mathrm{Sir}\:\mathrm{student}\:\mathrm{work},\:\mathrm{if}\:\mathrm{you}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{it}\:\mathrm{from} \\ $$$$\mathrm{there},\:\mathrm{better}\:\mathrm{start}\:\mathrm{learning}\:\mathrm{basic}\:\mathrm{integrals}. \\ $$$$\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{use}\:\mathrm{solving}\:\mathrm{problems}\:\mathrm{like}\:\mathrm{this}\:\mathrm{one} \\ $$$$\mathrm{for}\:\mathrm{you}\:\mathrm{if}\:\mathrm{you}\:\mathrm{cannot}\:\mathrm{understand}\:\mathrm{each}\:\mathrm{simple} \\ $$$$\mathrm{step}. \\ $$

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