Question

The value of $[{\int }_0^{\frac{\pi }{4}}\frac{2(1+x)\ln (1+x)\cdot \sin x+\cos x}{2(1+x){\cos }^{3}x}dx]$ is (where [.] denotes the greatest integer function)

Answer 1

$I={\int }_0^{\frac{\pi }{4}}\left(\frac{2\ln (1+x)\sin x\cos x+\frac{{\cos }^{2}x}{1+x}}{2{\cos }^{4}x}\right)dx{\int }_0^{\frac{\pi }{4}}\frac{1}{2}d\left(\frac{\ln (1+x)}{{\cos }^{2}x}\right)=\frac{1}{2}[\frac{\ln (1+x)}{{\cos }^{2}x}{]}_0^{\frac{\pi }{4}}=\ln (1+\frac{\pi }{4})$
So, [$I$] = 0