Category: Integration

Question Number 74218 by malikmasood3535@gmail.com last updated on 20/Nov/19 $$verifythaty\left(x\right)=e^x(\cos e^x-e^x\sin e^x)isthesolutionofintegralequationy\left(x\right)=(1-{xe}^{2x})\cos 1-e^{2x}\sin 1+\underset{0}{\stackrel{x}{\int }}\{1-(x-t)e^{2x}\}y\left(t\right)dt$$ Answered by mind is…

Question Number 74210 by malikmasood3535@gmail.com last updated on 20/Nov/19 $$\int e^t\cos e^{t}dt$$ Answered by mind is power last updated on 20/Nov/19 $$\int{e}^{{t}} {cos}\left({t}\right)\:{dt}={sin}\left({e}^{{t}}…

Question Number 8672 by swapnil last updated on 20/Oct/16 $$\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\mathrm{x}.{\mathrm{e}}^{-{\mathrm{x}}^2}\mathrm{dx}$$$$\mathrm{evaluate}\mathrm{above}\mathrm{expression}.$$ Answered by 123456 last updated on 21/Oct/16…

Question Number 139708 by qaz last updated on 30/Apr/21 $$Prove:1+2\underset{n=1}{\sum ^{\mathrm{\infty }}}{a}^n\cos \left(nx\right)=\frac{1-a^2}{1-2a\cos x+a^2},(\mid a\mid <1)$$$$Andcalculate::{\int }_0^{\pi }\frac{x^2}{1+8\sin {}^{2}x}dx=\frac{5{\pi }^3}{36}-\frac{\pi }{6}{ln}^{2}2$$…

Question Number 139696 by mnjuly1970 last updated on 30/Apr/21 Answered by Dwaipayan Shikari last updated on 30/Apr/21 $$\vartheta \left(\theta \right)=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{n+1}sin\left(n\theta \right)}{n}$$$$\vartheta\left(\theta\right)=\frac{\mathrm{1}}{\mathrm{2}{i}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}}…