If f(x)=7tan^8x+7tan^6x-3tan^4x-3tan^2x if I1=∫ f(x)dx, I2 x ∈ to (0, π/4)

Question

If \(f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x\) if \(I_1=\int_0^{\frac{\pi}{4}} f(x) d x, I_2=\int_0^{\frac{\pi}{4}} x f(x) d x\) then find \(7 I_1+12 I_2\)

Answer 1

Answer is "1" 

\(f(x)=\left(7 \tan ^6 x-3 \tan ^2 x\right) \sec ^2 x\)  

\(I_1=\int_0^{\frac{\pi}{4}}\left(7 \tan ^6 x-3 \tan ^2 x\right) \sec ^2 x d x\)     

Let \(\tan \mathrm{x}=\mathrm{t}\) 

\(=\int_0^1\left(7 t^6-3 t^4\right) d t=\left.\left(t^7-t^3\right)\right|_0 ^1=0\)