Question

${\int }_0^{4\pi }co{s}^{2014}xdx$ is equal to

• A. $\frac{\mathrm{4.2013.2011.2009.........3.1}}{\mathrm{2014.2012.2010.......4.2}}$
• B. $\frac{\mathrm{2.2013.2011.2009.........3.1.}\pi }{\mathrm{2014.2012.2010........6.4}}$
• C. $\frac{\mathrm{4.1013.2011.2009.......3.1.}\pi }{\mathrm{2014.2012.2010..........6.4}}$
• D. $\frac{\mathrm{8.2013.2011.2009........3.1}}{\mathrm{2014.2012.2010...........4.2}}$

Answer 1

Answer: (B)

$\frac{\mathrm{2.2013.2011.2009.........3.1.}\pi }{\mathrm{2014.2012.2010........6.4}}$

${\int }_0^{4\pi }co{s}^{2014}xdx$

$=8{\int }_0^{\frac{\pi }{2}}co{s}^{2014}xdx$

$=8\times \left(\frac{2013}{2014}\right)\left(\frac{2011}{2012}\right)\left(\frac{2009}{2010}\right).......\left(\frac{3}{4}\right)\left(\frac{1}{2}\right)\cdot \frac{\pi }{2}$

$=\frac{\mathrm{2.2013.2011.2009.......3.1.}\pi }{\mathrm{2014.2012.2010......6.4}}$