Question Number 21582 by Isse last updated on 28/Sep/17
$$\int_{\pi/\mathrm{2}} ^{\pi/\mathrm{4}} \left(\mathrm{3}{x}+\mathrm{7}\right) \\ $$
Answered by Joel577 last updated on 29/Sep/17
![I = ∫_(π/2) ^(π/4) 3x + 7 dx = [(3/2)x^2 + 7x]_(π/2) ^(π/4) = (((3π^2 )/(32)) + ((7π)/4)) − (((3π^2 )/8) + ((7π)/2)) = −(((9π^2 )/(32)) + ((7π)/4))](https://www.tinkutara.com/question/Q21589.png)
$${I}\:=\:\underset{\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\:\mathrm{3}{x}\:+\:\mathrm{7}\:{dx} \\ $$$$\:\:\:=\:\left[\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} \:+\:\mathrm{7}{x}\right]_{\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{4}}} \\ $$$$\:\:\:=\:\left(\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{32}}\:+\:\frac{\mathrm{7}\pi}{\mathrm{4}}\right)\:−\:\left(\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{8}}\:+\:\frac{\mathrm{7}\pi}{\mathrm{2}}\right) \\ $$$$\:\:\:=\:−\left(\frac{\mathrm{9}\pi^{\mathrm{2}} }{\mathrm{32}}\:+\:\frac{\mathrm{7}\pi}{\mathrm{4}}\right) \\ $$