Category: Integration

Question Number 83675 by niroj last updated on 05/Mar/20 $$$$$$$$$$\mathrm{evaluate}:$$$$2{\int }_0^2\frac{\sqrt{\mathrm{x}+1}}{{\mathrm{x}}^2+4}\mathrm{dx}$$$$$$$$$$…

Question Number 149157 by gsk2684 last updated on 03/Aug/21 $$\underset{\frac{1}{2}}{\stackrel{2}{\int }}\frac{1}{x}\mathrm{cosec}{}^{101}(x-\frac{1}{x})dx=?$$ Answered by Kamel last updated on 03/Aug/21 $$0$$…

Question Number 18066 by tawa tawa last updated on 14/Jul/17 $$\left(\mathrm{a}\right)\mathrm{Evaluate}\mathrm{the}\mathrm{integral}\mathrm{of}\mathrm{the}\mathrm{function}:\mathrm{y}\left(\mathrm{x}\right)=\frac{3\mathrm{x}+1}{{2\mathrm{x}}^2-2\mathrm{x}+3}$$$$\left(\mathrm{b}\right)\mathrm{Find}\mathrm{the}\mathrm{constant}\mathrm{A},\mathrm{B},\mathrm{C}\mathrm{in}\mathrm{the}\mathrm{identity}:$$$$\frac{{3\mathrm{x}}^2-\mathrm{ax}}{(\mathrm{x}-2\mathrm{a})({\mathrm{x}}^2+{\mathrm{a}}^2)}\equiv \frac{\mathrm{A}}{(\mathrm{x}-2\mathrm{a})}+\frac{\mathrm{Bx}+\mathrm{Ca}}{({\mathrm{x}}^2+{\mathrm{a}}^2)}$$$$\mathrm{where}\mathrm{a}\mathrm{is}\mathrm{a}\mathrm{constant},\mathrm{hence}\mathrm{prove}\mathrm{that}.$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}}…

Question Number 83597 by dennis kenneth last updated on 04/Mar/20 $${\int }_0^2(3x^2-4x+2)dx$$ Commented by Jidda28 last updated on 04/Mar/20 $$\left.=\:\:{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}}…

Question Number 149112 by bramlexs22 last updated on 03/Aug/21 $$\phi =\int \tan (\mathrm{x}+\frac{\pi }{3})\tan 3\mathrm{x}\tan (2\mathrm{x}-\frac{\pi }{3})\mathrm{dx}=?$$ Commented by som(math1967) last updated on 03/Aug/21 $$\mathit{t}\mathit{a}\mathit{n}3\mathit{x}=\mathit{t}\mathit{a}\mathit{n}(\mathit{x}+\frac{\mathit{\pi }}{3}+2\mathit{x}-\frac{\mathit{\pi }}{3})$$$$\mathit{t}\mathit{a}\mathit{n}3\mathit{x}=\frac{\mathit{t}\mathit{a}\mathit{n}(\mathit{x}+\frac{\mathit{\pi }}{3})+\mathit{t}\mathit{a}\mathit{n}(2\mathit{x}-\frac{\mathit{\pi }}{3})}{1-tan(\mathit{x}+\frac{\mathit{\pi }}{3})\mathit{t}\mathit{a}\mathit{n}(2\mathit{x}+\frac{2\mathit{\pi }}{3})}$$$$\therefore\boldsymbol{{tan}}\mathrm{3}\boldsymbol{{xtan}}\left(\boldsymbol{{x}}+\frac{\boldsymbol{\pi}}{\mathrm{3}}\right)\boldsymbol{{tan}}\left(\mathrm{2}\boldsymbol{{x}}−\frac{\boldsymbol{\pi}}{\mathrm{3}}\right) \…