Category: Integration

Question Number 188384 by universe last updated on 28/Feb/23 $$$$$$\mathrm{if}{\int }_0^{\mathrm{\infty }}{e}^{-ax}dx=\frac{1}{a}$$$$showthat{\int }_0^{\mathrm{\infty }}{e}^{-ax}{x}^{n}dx=\frac{n!}{a^{n+1}}$$ Answered…

Question Number 188380 by Shlock last updated on 28/Feb/23 Answered by SEKRET last updated on 28/Feb/23 $$\mathbf{f}\left(\mathbf{t}\right)={\int }_0^{\mathit{\pi }}\frac{\mathbf{ln}(1+\mathbf{t}\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right))}{\mathbf{cos}\left(\mathbf{x}\right)}\mathbf{dx}$$$$\mathbf{f}{}^{\prime }\left(\mathbf{t}\right)={\int }_0^{\mathit{\pi }}\frac{\mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right)}{\mathbf{cos}\left(\mathbf{x}\right)\cdot (1+\mathbf{t}\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right))}\mathbf{dx}$$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\int_{\mathrm{0}}…

Question Number 122828 by bemath last updated on 20/Nov/20 Answered by bobhans last updated on 20/Nov/20 $$solve\underset{\pi /4}{\stackrel{\pi /3}{\int }}\frac{\sqrt{\tan x}}{\sin 2x}dx.$$$$Solution:$$$$B\left(x\right)=\int \frac{\sqrt{\tan x}}{2\sin x\cos x}dx$$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:{x}}\:\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:{x}}}…

Question Number 122822 by bemath last updated on 19/Nov/20 Answered by liberty last updated on 20/Nov/20 $$I=\int x\tan x\sec {}^{2}xdx=\int x\tan xd\left(\tan x\right)$$$$byD.Imethod\to \{\begin{array}{l}u=x\to du=dx\\ v=\int \tan xd\left(\tan x\right)=\frac{1}{2}\tan {}^{2}x\end{array}$$$${I}=\frac{\mathrm{1}}{\mathrm{2}}{x}.\mathrm{tan}\:^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{2}}\int\mathrm{tan}\:^{\mathrm{2}} {x}\:{dx}\:…

Question Number 57241 by Aditya789 last updated on 01/Apr/19 $$\int \frac{\times \sqrt{\mathrm{x}+1}}{\mathrm{x}+2}\mathrm{dx}$$ Answered by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19 $$x+1=t^2\to dx=2tdt$$$$\int\frac{\left({t}^{\mathrm{2}} −\mathrm{1}\right){t}×\mathrm{2}{tdt}}{{t}^{\mathrm{2}} +\mathrm{1}}…