If _( )/(4)^0 ^2x1+ x xdx=(1)/(a) _e((a)/(3))+ b√3, where a,b N, then a+b is equal to ____…

Question

If ${\int }_{\frac{\pi }{4}}^{0}\frac{{\sin }^{2}x}{1+\sin x\cos x}dx=\frac{1}{a}{\log }_{e}(\frac{a}{3})+\frac{\pi }{b\sqrt{3}}$, where $a,b\in N$, then $a+b$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 8

Accepted Answer

$I=\int _{0}^{\frac{\pi }{4}}\frac{{\sin }^{2}x}{1+\sin x\cos x}dx=\int _{0}^{\frac{\pi }{4}}\frac{{\sin }^{2}x}{{\sin }^{2}x+{\cos }^{2}x+\sin x\cos x}dx$
$I=\int _{0}^{\frac{\pi }{4}}\frac{{	an }^{2}x}{1+	an x+{	an }^{2}x}dx$

$=\int _{0}^{\frac{\pi }{4}}\frac{	an x\cdot {\sec }^{2}xdx}{(1+{	an }^{2}x)(1+	an x+{	an }^{2}x)}$

Let $	an x=t$

$\begin{gathered} I =\int _{0}^{1}\frac{{t}^{2}}{(1+{t}^{2})(1+t+{t}^{2})}dt \ =\int _{0}^{1}(\frac{x}{1+{x}^{2}}-\frac{x}{1+x+{x}^{2}})dx \end{gathered}$

$\frac{1}{2}\int _{0}^{1}\frac{2x}{1+{x}^{2}}dx-\int _{0}^{}\frac{\frac{1}{2}(2x+1)-\frac{1}{2}}{1+x+{x}^{2}}dx$

$\begin{gathered} =\frac{1}{2}\ln 2-\frac{1}{2}\ln 3\frac{1}{2}\int _{0}^{1}\frac{dx}{{(x+\frac{1}{2})}^{2}+\frac{3}{4}} \ =\frac{1}{2}\ln \frac{2}{3}+\frac{1}{2}\cdot \frac{2}{\sqrt{3}}{[{	an }^{-1}\frac{2x+1}{\sqrt{3}}]}_{1}^{0} \ =\frac{1}{2}\ln \frac{2}{3}+\frac{1}{\sqrt{3}}(\frac{\pi }{3}-\frac{\pi }{6}) \ =\frac{1}{2}\ln \frac{2}{3}+\frac{1}{\sqrt{3}}\cdot \frac{\pi }{6} \ 	herefore \, a=2,b=6 \ 	herefore \, a+b=8 \end{gathered}$

Solutions · 1