Category: Integration

Question Number 206639 by mathlove last updated on 21/Apr/24 Commented by Frix last updated on 21/Apr/24 $$=\underset{\frac{\pi }{12}}{\stackrel{\frac{\pi }{11}}{\int }}(\frac{\sqrt{s}}{2(\sqrt{c}+\sqrt{s})}-\frac{\sqrt{cs}}{c+\sqrt{s}(\sqrt{c}+1)}+\frac{(\sqrt{c}-1)s+\sqrt{cs}}{2((\sqrt{s}+1)c+(\sqrt{c}+1)s)})dx$$$$\mathrm{With}c=\cos x\wedge s=\sin x$$$$\mathrm{I}\mathrm{doubt}\mathrm{we}\mathrm{can}\mathrm{solve}\mathrm{this}.$$…

Question Number 206579 by York12 last updated on 19/Apr/24 $$\mathrm{Evaluate}:\underset{0}{\stackrel{1}{\int }}\frac{\ln (1+x^2)}{1+x}d\left(x\right).$$ Answered by Berbere last updated on 19/Apr/24 $$=\left[{ln}\left(\mathrm{1}+{x}\right){ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right]_{\mathrm{0}} ^{\mathrm{1}}…

Question Number 206557 by luciferit last updated on 18/Apr/24 Answered by Frix last updated on 18/Apr/24 $$\mathrm{Without}\mathrm{substitution}:$$$$\int \frac{3+{2\sin }^{2}x}{{\cos }^{2}x}dx=\int (\frac{5}{{\cos }^{2}x}-2)dx=$$$$=\mathrm{5tan}\:{x}\:−\mathrm{2}{x}\:+{C} \…

Question Number 206541 by luciferit last updated on 18/Apr/24 Answered by lepuissantcedricjunior last updated on 18/Apr/24 $$\int \frac{3\mathit{x}+5}{\sqrt{{\mathit{x}}^2-10\mathit{x}+51}}\mathit{d}\mathit{x}=\mathit{k}$$$$\mathit{k}=\frac{3}{2}\int \frac{2\mathit{x}-5+\frac{10}{3}+5}{\sqrt{{\mathit{x}}^2-10\mathit{x}+51}}\mathit{d}\mathit{x}$$$$\boldsymbol{{k}}=\mathrm{3}\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{10}\boldsymbol{{x}}+\mathrm{51}}+\frac{\mathrm{25}}{\mathrm{2}}\int\frac{\boldsymbol{{dx}}}{\:\sqrt{\left(\boldsymbol{{x}}−\mathrm{5}\right)^{\mathrm{2}} +\mathrm{26}}}…

Question Number 206542 by luciferit last updated on 18/Apr/24 Answered by lepuissantcedricjunior last updated on 18/Apr/24 $$\int (3\mathit{x}+5)\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\left(\mathit{x}\right)\mathit{d}\mathit{x}=\mathit{k}$$$$\{\begin{array}{l}\mathit{u}=\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\left(\mathit{x}\right)\\ {\mathit{v}}^{\prime }=(3\mathit{x}+5)\end{array}=>\{\begin{array}{l}{\mathit{u}}^{\prime }=\frac{1}{1+{\mathit{x}}^2}\\ \mathit{v}=\frac{3}{2}{\mathit{x}}^2+5\mathit{x}\end{array}$$$$\boldsymbol{{k}}=\left(\frac{\mathrm{3}}{\mathrm{2}}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\boldsymbol{{x}}\right)\boldsymbol{{arctan}}\left(\boldsymbol{{x}}\right)−\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}+\frac{\mathrm{10}}{\mathrm{3}}\boldsymbol{{x}}−\mathrm{1}}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}}…