Question Number 224505 by EmGent last updated on 15/Sep/25 $$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{{ax}} {J}_{\mathrm{0}} \left(\jmath_{\mathrm{0}{m}} {x}\right){dx} \\ $$$$\mathrm{Where}\:{J}_{\mathrm{0}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{and}\:\jmath_{\mathrm{0}{m}} \:\mathrm{its}\:{m}-\mathrm{th}\:\mathrm{zero} \\ $$ Terms of Service Privacy…
Question Number 224373 by Nicholas666 last updated on 05/Sep/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:; \\ $$$$\:\:\:\:\:\mathcal{I}\:=\:\underset{\:\:\mathrm{0}} {\overset{\:\:\mathrm{1}} {\int}}\underset{\:\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\:\frac{\mathrm{ln}\left(\mathrm{1}+\sqrt{{xy}}\right)\:\mathrm{ln}\left(\mathrm{1}+\:\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}−{y}}}\right)}{\:\sqrt{\mathrm{1}−{x}}\:\:\sqrt{\mathrm{1}−{y}}\:\:\left({x}+{y}\right)}\:\:{dxdy}\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\mathcal{I}\:=\:\zeta\left(\mathrm{3}\right)−\frac{\mathrm{70}}{\mathrm{351}}−\frac{\mathrm{280}}{\mathrm{351}}\:\mathrm{ln}\:\mathrm{2}−\frac{\mathrm{40}}{\mathrm{117}}\:\mathrm{ln}^{\mathrm{2}} \:\mathrm{2}\:+\frac{\mathrm{412}}{\mathrm{351}}\:\mathrm{ln}^{\mathrm{3}} \:\mathrm{2}\:+\:\frac{\mathrm{167}}{\mathrm{2106}}\:\pi^{\mathrm{2}} \:\mathrm{ln}\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$…
Question Number 224248 by Nicholas666 last updated on 29/Aug/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}−\sqrt{\mathrm{x}}\right)}{\mathrm{x}}\:\:\mathrm{dx} \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 224201 by MirHasibulHossain last updated on 24/Aug/25 $$\int\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{7}} −\mathrm{8x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 224182 by CrispyXYZ last updated on 24/Aug/25 $$\mathrm{Calculate} \\ $$$${I}=\int\:\frac{\mathrm{sin}\:{x}}{\mathrm{1}+\:\mathrm{sin}\:{x}}\:\mathrm{d}{x} \\ $$ Answered by Frix last updated on 24/Aug/25 $$\int\frac{\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{sin}\:{x}}{dx}=\int\frac{\left(\mathrm{1}−\mathrm{sin}\:{x}\right)\mathrm{sin}\:{x}}{\left(\mathrm{1}−\mathrm{sin}\:{x}\right)\left(\mathrm{1}+\mathrm{sin}\:{x}\right)}{dx}= \\ $$$$=\int\frac{\mathrm{sin}\:{x}\:−\mathrm{sin}^{\mathrm{2}} \:{x}}{\mathrm{cos}^{\mathrm{2}}…
Question Number 224085 by MathematicalUser2357 last updated on 18/Aug/25 $$\mathrm{If}\:{f}\left({x}\right)=\mathrm{4}{x}^{\mathrm{3}} +\mathrm{3}{x}^{\mathrm{2}} +{x},\:\mathrm{Then}\:\mathrm{solve}\:\mathrm{for}\:{a}\:\mathrm{and}\:{b}: \\ $$$$\underset{{x}\in\mathbb{R}} {\mathrm{max}}\left\{\int_{{x}} ^{\mathrm{2}} {f}\left({t}\right){dt}\right\}={a}\:\mathrm{where}\:{x}={b} \\ $$ Answered by Ghisom_ last updated on…
Question Number 223958 by Nicholas666 last updated on 11/Aug/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}\:\mathrm{sinh}\left({x}\right)}{\mathrm{1}+\mathrm{cosh}^{\mathrm{2}} \left({x}\right)}\:\mathrm{d}{x} \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 223933 by MathematicalUser2357 last updated on 10/Aug/25 $${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left({x}+\mathrm{1}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$ Answered by Tawa11 last updated on 10/Aug/25 $$\mathrm{Note}:\:\:\:\:\mathrm{I}\left(\mathrm{a}\right)\:\:=\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{a}} \:\frac{\mathrm{ln}\left(\mathrm{1}\:\:+\:\:\mathrm{ax}\right)}{\mathrm{1}\:\:+\:\:\mathrm{x}^{\mathrm{2}}…
Question Number 223920 by Mathspace last updated on 09/Aug/25 $${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}} }{{sin}^{\mathrm{2}} {x}}{dx} \\ $$ Answered by Frix last updated on 09/Aug/25 $$\mathrm{by}\:\mathrm{parts}\:\left(\mathrm{2}\:\mathrm{times}\right) \\…
Question Number 223923 by Mathspace last updated on 09/Aug/25 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$ Answered by Tawa11 last updated on 14/Aug/25 $$\:\mathrm{Finally}:\: \\ $$$$\:\:\:\:\int_{\:\mathrm{0}}…