Question Number 65805 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:\:{Prove}\:{that}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\mathrm{1}+\left({tant}\right)^{{n}} }\:\:{does}\:{not}\:{depend}\:{of}\:{the}\:{term}\:{n} \\ $$$${deduces}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2035}}…
Question Number 261 by raj last updated on 25/Jan/15 $$\mathrm{If}\:\underset{\mathrm{0}} {\overset{{x}} {\int}}{f}\left({t}\right){dt}={x}+\underset{{x}} {\overset{\mathrm{1}} {\int}}{tf}\left({t}\right){dt},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{f}\left(\mathrm{1}\right). \\ $$ Answered by prakash jain last updated on…
Question Number 260 by 9999 last updated on 25/Jan/15 $$\int_{−\mathrm{1}} ^{+\mathrm{1}} \mid\mathrm{1}−{x}\mid{dx}= \\ $$ Answered by 123456 last updated on 17/Dec/14 $$\mid\mathrm{1}−{x}\mid=\begin{cases}{\mathrm{1}−{x}}&{{x}\leqslant\mathrm{1}}\\{{x}−\mathrm{1}}&{{x}\geqslant\mathrm{1}}\end{cases} \\ $$$$\int_{−\mathrm{1}} ^{+\mathrm{1}}…
Question Number 259 by a@b.c last updated on 25/Jan/15 $$\int_{{a}} ^{{b}} \:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx}= \\ $$ Answered by prakash jain last updated on 17/Dec/14 $$\mathrm{Substitue}\:{x}={a}+{b}−{y}\:\Rightarrow{dx}=−{dy} \\ $$$$\mathrm{The}\:\mathrm{given}\:\mathrm{integral}\:{I}…
Question Number 65788 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $${Explicit}\:\:\:{f}\left({a}.{b}.{c}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{sec}\left({x}−{a}\right)}{{b}.{cosx}\:+\:{c}.{sinx}}\:{dx} \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 65786 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:{Shows}\:{that}\:\:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} =\frac{\pi}{{xsinh}\left(\pi{x}\right)}\:\:\:\:\:\:{with}\:\Gamma\left({z}\right)=\int_{\mathrm{0}_{} } ^{\infty} \:{t}^{{z}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$${Then}\:{Prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}}…
Question Number 243 by 123456 last updated on 25/Jan/15 $$\mathrm{evaluate} \\ $$$$\underset{−\infty} {\overset{+\infty} {\int}}\frac{\mathrm{sin}\:{x}}{{x}}{dx} \\ $$ Answered by prakash jain last updated on 17/Dec/14 $$\mathrm{Let}\:\mathrm{us}\:\mathrm{consider}\:\underset{\mathrm{0}}…
Question Number 65776 by mathmax by abdo last updated on 03/Aug/19 $${find}\:\:\int_{−\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{cosx}}{\mathrm{2}+\mathrm{5}{sinx}}{dx} \\ $$ Commented by kaivan.ahmadi last updated on 04/Aug/19 $${u}=\mathrm{2}+\mathrm{5}{sinx}\Rightarrow{du}=\mathrm{5}{cosxdx} \\…
Question Number 65774 by mathmax by abdo last updated on 03/Aug/19 $${find}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{sin}^{\mathrm{2}} {x}}{{sin}^{\mathrm{2}} \left(\frac{{nx}}{\mathrm{2}}\right)}{dx}\:\:\:\:\left({n}>\mathrm{0}\right) \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 65773 by mathmax by abdo last updated on 03/Aug/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{1}+{x}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){detemine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{\left(\mathrm{1}+{x}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)^{\mathrm{2}} }{dt} \\…