Question Number 543 by 123456 last updated on 25/Jan/15 $${if}\:{f}'\left({x}\right)={f}\left({x}\right)+{x},\:{f}\left(\mathrm{0}\right)=\mathrm{0},\:{then} \\ $$$${f}\left(\mathrm{1}\right)=? \\ $$ Answered by prakash jain last updated on 24/Jan/15 $${y}={f}\left({x}\right) \\ $$$${y}'−{y}={x}…
Question Number 542 by 123456 last updated on 25/Jan/15 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} }{\:\sqrt{{x}^{\mathrm{6}} +\mathrm{4}}}{dx} \\ $$ Answered by prakash jain last updated on 24/Jan/15 $${x}^{\mathrm{3}}…
Question Number 131614 by mnjuly1970 last updated on 06/Feb/21 $$\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}\::\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){cos}\left(\mathrm{2}\pi{nx}\right){dx}=\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$$${for}\:{example}\::\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){cos}\left(\mathrm{2}\pi{x}\right){dx}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$ Terms of Service…
Question Number 537 by kth last updated on 25/Jan/15 $$\int_{\mathrm{4}} ^{\mathrm{5}} \frac{{x}−\mathrm{2}}{\left({x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}\right)^{\mathrm{3}} }{dx} \\ $$ Answered by Vishal last updated on 25/Jan/15 $${let}\:{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}\:=\:{t}\:{then}\:\left({x}−\mathrm{2}\right)\:{dx}\:=\:\frac{{dt}}{\mathrm{2}}…
Question Number 536 by kth last updated on 25/Jan/15 $$\int_{\mathrm{1}} ^{\mathrm{4}} \frac{\mathrm{2}}{\:\sqrt{{x}}\left(\sqrt{{x}}+\mathrm{4}\right)^{\mathrm{3}} }{dx} \\ $$$$ \\ $$ Answered by Vishal last updated on 24/Jan/15 $${let}\:\sqrt{{x}}+\mathrm{4}={t}\:{then}\:\frac{\mathrm{1}}{\:\sqrt{{x}}}\:{dx}=\mathrm{2}\:{dt}…
Question Number 131605 by physicstutes last updated on 06/Feb/21 $$\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:{m}^{\mathrm{4}} −\mathrm{7}{m}^{\mathrm{3}} +\:\mathrm{14}{m}^{\mathrm{2}} −\mathrm{7}{m}\:+\:\mathrm{1}\:=\:\mathrm{0}\: \\ $$ Answered by malwan last updated on 06/Feb/21 $${m}^{\mathrm{4}}…
Question Number 535 by kth last updated on 25/Jan/15 $$\int_{\mathrm{4}} ^{\mathrm{1}} \frac{\mathrm{2}}{\:\sqrt{{x}}} \\ $$ Answered by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$=\mathrm{2}\int_{\mathrm{4}} ^{\mathrm{1}} {x}^{\frac{−\mathrm{1}}{\mathrm{2}}} {dx}…
Question Number 66071 by AnjanDey last updated on 08/Aug/19 $$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{as}\:\mathrm{a}\:\mathrm{limit}\:\mathrm{of}\:\mathrm{sums}: \\ $$$$\mathrm{1}.\int_{\mathrm{1}} ^{\mathrm{3}} \left({e}^{\mathrm{2}−\mathrm{3}{x}} +{x}^{\mathrm{2}} +\mathrm{1}\right){dx} \\ $$ Answered by meme last updated on 08/Aug/19…
Question Number 532 by 123456 last updated on 25/Jan/15 $${if}\:{f}\:{is}\:{continuos}\:{and}\:{diferentiable} \\ $$$${everywhere}\:{on}\:\mathbb{R},\:{if}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{and} \\ $$$$\mid{f}'\left({x}\right)\mid\leqslant\mid{f}\left({x}\right)\mid\:{then}\:{proof}\:{that} \\ $$$${f}\left({x}\right)=\mathrm{0} \\ $$ Answered by prakash jain last updated on…
Question Number 66066 by Kunal12588 last updated on 08/Aug/19 $${Integrate}\:\int\frac{{dx}}{{ax}^{\mathrm{2}} +{bx}+{c}} \\ $$$${y}\:=\:{ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${y}'=\frac{{dy}}{{dx}}=\mathrm{2}{ax}+{b} \\ $$$${d}\:=\:\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}} \\ $$$${d}'\:=\:\sqrt{−{d}} \\ $$$${Case}\:\mathrm{1}.\:{d}^{\mathrm{2}} \:<\:\mathrm{0} \\…