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Category: Integration

prove-that-1-0-lt-0-pi-4-x-tan-x-dx-lt-pi-2-32-2-1-2-lt-pi-4-pi-2-sin-x-x-dx-lt-2-2-3-0-lt-100pi-200pi-cos-x-x-dx-lt-1-100pi-

Question Number 12826 by malwaan last updated on 03/May/17 $$\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{1}:\:\:\mathrm{0}<\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\mathrm{x}\sqrt{\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}<\:\frac{\pi^{\mathrm{2}} }{\mathrm{32}} \\ $$$$\mathrm{2}:\:\frac{\mathrm{1}}{\mathrm{2}}<\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\:\mathrm{dx}\:<\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\mathrm{3}:\:\mathrm{0}<\int_{\mathrm{100}\pi} ^{\mathrm{200}\pi} \:\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{x}}\:\mathrm{dx}\:<\frac{\mathrm{1}}{\mathrm{100}\pi} \\ $$…

0-pi-2-cos-2-cos-x-sin-2-sin-x-dx-

Question Number 78334 by john santu last updated on 16/Jan/20 $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\left[\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{cos}\:{x}\right)+\mathrm{sin}^{\mathrm{2}} \:\left(\mathrm{sin}\:{x}\right)\:\right]\:{dx} \\ $$ Commented by MJS last updated on 16/Jan/20 $$\mathrm{we}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{integral}\:\mathrm{but}\:\mathrm{we}\:\mathrm{can}…

dx-1-tan-x-

Question Number 12766 by Joel577 last updated on 01/May/17 $$\int\:\frac{{dx}}{\mathrm{1}\:+\:\mathrm{tan}\:{x}} \\ $$ Answered by sma3l2996 last updated on 01/May/17 $${t}={tanx}\Rightarrow{dt}=\left(\mathrm{1}+{tan}^{\mathrm{2}} {x}\right){dx} \\ $$$${dx}=\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\…

sec-x-tan-2-x-dx-

Question Number 12768 by Joel577 last updated on 01/May/17 $$\int\:\frac{\mathrm{sec}\:{x}}{\mathrm{tan}^{\mathrm{2}} \:{x}}\:{dx} \\ $$ Answered by prakash jain last updated on 01/May/17 $$\int\:\frac{\mathrm{sec}\:{x}\centerdot\mathrm{cos}^{\mathrm{2}} {x}}{\mathrm{sin}^{\mathrm{2}} {x}}\:\mathrm{d}{x} \\…