Question Number 150037 by RoswelCod2003 last updated on 09/Aug/21
![Random Problem: ∫_(π/4) ^(π/2) (−7sin x + 3cos x) dx By getting the antiderivative of the trigonometric functions: ∫ sin(x) dx = −cos x + c ∫ cos(x) dx = sin x + c = −7 ∫ sin x + 3 ∫ cos x ∣_(π/4) ^(π/2) = −7(− cos x) + 3(sin x) ∣_(π/4) ^(π/2) = 7 cos x + 3sin x ∣_(π/4) ^(π/2) Evaluate it to the top and bottom limit of integration: = (7 cos ∙ (π/2) + 3 sin ∙ (π/2))− (7 cos ∙ (π/(4 )) + 3 sin ∙ (π/4) ) =[7(0) + 3(1)] − [7(((√2)/2)) + 3(((√2)/2))] = 3 − ((7(√2))/2) − ((3(√2))/2) = 3 − ((10(√2))/2) or 3 − 5(√2) Answer: 3 − 5(√2) Solution by Roswel:)](Q150037.png)
$${Random}\:{Problem}: \\ $$$$\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\left(−\mathrm{7sin}\:{x}\:+\:\mathrm{3cos}\:{x}\right)\:{dx} \\ $$$$ \\ $$$${By}\:{getting}\:{the}\:{antiderivative}\:{of}\:{the}\:{trigonometric}\:{functions}: \\ $$$$\int\:\mathrm{sin}\left({x}\right)\:{dx}\:=\:−\mathrm{cos}\:{x}\:+\:{c} \\ $$$$\int\:\mathrm{cos}\left({x}\right)\:{dx}\:=\:\mathrm{sin}\:{x}\:+\:{c} \\ $$$$=\:−\mathrm{7}\:\int\:\mathrm{sin}\:{x}\:\:+\:\:\mathrm{3}\:\int\:\mathrm{cos}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}}\:=\:−\mathrm{7}\left(−\:\mathrm{cos}\:{x}\right)\:+\:\mathrm{3}\left(\mathrm{sin}\:{x}\right)\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}} \\ $$$$=\:\mathrm{7}\:\mathrm{cos}\:{x}\:+\:\mathrm{3sin}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}} \\ $$$$ \\ $$$${Evaluate}\:{it}\:{to}\:{the}\:{top}\:{and}\:{bottom}\:{limit}\:{of}\:{integration}: \\ $$$$ \\ $$$$=\:\left(\mathrm{7}\:\mathrm{cos}\:\centerdot\:\frac{\pi}{\mathrm{2}}\:+\:\mathrm{3}\:\mathrm{sin}\:\centerdot\:\frac{\pi}{\mathrm{2}}\right)−\:\left(\mathrm{7}\:\mathrm{cos}\:\centerdot\:\frac{\pi}{\mathrm{4}\:}\:\:+\:\mathrm{3}\:\mathrm{sin}\:\centerdot\:\frac{\pi}{\mathrm{4}}\:\right) \\ $$$$=\left[\mathrm{7}\left(\mathrm{0}\right)\:+\:\mathrm{3}\left(\mathrm{1}\right)\right]\:−\:\left[\mathrm{7}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\:+\:\mathrm{3}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\right] \\ $$$$=\:\mathrm{3}\:−\:\frac{\mathrm{7}\sqrt{\mathrm{2}}}{\mathrm{2}}\:−\:\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$=\:\mathrm{3}\:−\:\frac{\mathrm{10}\sqrt{\mathrm{2}}}{\mathrm{2}}\:{or}\:\mathrm{3}\:−\:\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$${Answer}:\:\mathrm{3}\:−\:\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$\left.{Solution}\:{by}\:{Roswel}:\right) \\ $$