Category: Integration

Question Number 126986 by mnjuly1970 last updated on 25/Dec/20 $$\dots nicecalculus\dots$$$$provethat::$$$$\mathrm{I}:={\int }_0^{\frac{\pi }{2}}\frac{\left\{cot\left(x\right)\right\}}{cot\left(x\right)}dx=\frac{1}{2}(\pi -ln\left(\frac{sinh\left(\pi \right)}{\pi }\right))$$$$\left\{x\right\}isfractionalpartofx..$$ Answered by Olaf last updated…

Question Number 192470 by Spillover last updated on 19/May/23 Answered by Spillover last updated on 19/May/23 $${\int }_0^{\sqrt{2}}\sqrt{1+{\left(2x\right)}^2}dx$$$$Let2x=\sinh \theta dx=\frac{\cosh \theta d\theta }{2}$$$$\int_{\mathrm{0}} ^{\sqrt{\mathrm{2}}}…

Question Number 61386 by maxmathsup by imad last updated on 02/Jun/19 $$findthevalueof{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{ln(1+x^2)}{1+x^2}dx$$ Commented by perlman last updated on 02/Jun/19…

Question Number 192452 by Spillover last updated on 18/May/23 $$Evaluate{\int }_0^1{2}^{x}dx$$$$$$ Answered by senestro last updated on 18/May/23 $$\mathrm{1}/\mathrm{ln}\:\mathrm{2}…

Question Number 61349 by tanmay last updated on 01/Jun/19 Commented by MJS last updated on 01/Jun/19 $$\dots \mathrm{harder}\mathrm{than}\mathrm{I}\mathrm{thought}$$$$\int \frac{\sqrt{t^2-5}}{t-a}dt=$$$$[t=\frac{\sqrt{5}}{\sin u}\iff u=\arcsin \frac{\sqrt{5}}{t}\to dt=-\frac{t\sqrt{t^2-5}}{\sqrt{5}}du]$$$$\:\:\:\:\:\mathrm{now}\:\mathrm{the}\:\mathrm{root}\:\mathrm{is}\:\mathrm{gone},\:\mathrm{but}\:\mathrm{we}\:\mathrm{must}\:\mathrm{make}…