Question

the value of integration (0-π|2)(dx/1+tan^3x)

Answer 1

$\text{Let }I=\int\limits^{\frac{\pi}{2}}_0\;\frac{1}{1+tan^3x}\;dx$

$\implies\;I=\int\limits^{\frac{\pi}{2}}_0\;\frac{1}{1+\frac{sin^3x}{cos^3x}}\;dx$

$\implies\;I=\int\limits^{\frac{\pi}{2}}_0\;\frac{cos^3x}{cos^3x+sin^3x}\;dx$...........(1)

$\text{Using the property}$

$\boxed{\bf\int\limits^a_0{f(x)}\,dx=\int\limits^a_0{f(a-x)}\,dx}$

$\implies\;I=\int\limits^{\frac{\pi}{2}}_0\;\frac{cos^3(\frac{\pi}{2}-x)}{cos^3(\frac{\pi}{2}-x)+sin^3(\frac{\pi}{2}-x)}\;dx$

$\implies\;I=\int\limits^{\frac{\pi}{2}}_0\;\frac{sin^3x}{sin^3x+cos^3x}\;dx$

$\implies\;I=\int\limits^{\frac{\pi}{2}}_0\;\frac{sin^3x}{cos^3x+sin^3x}\;dx$..........(2)

$\text{Adding (1) and (2), we get}$

$2I=\int\limits^{\frac{\pi}{2}}_0\;\frac{cos^3x}{cos^3x+sin^3x}\;dx+\int\limits^{\frac{\pi}{2}}_0\;\frac{sin^3x}{cos^3x+sin^3x}\;dx$

$\implies\,2I=\int\limits^{\frac{\pi}{2}}_0\;\frac{cos^3x+sin^3x}{cos^3x+sin^3x}\;dx$

$\implies\,2I=\int\limits^{\frac{\pi}{2}}_0\;dx$

$\implies\,2I=[x]\limits^{\frac{\pi}{2}}_0$

$\implies\,2I=\frac{\pi}{2}-0$

$\implies\,2I=\frac{\pi}{2}$

$\implies\,\boxed{\bf\,I=\frac{\pi}{4}}$