I =∫& lt; 0,π& gt; (sinx)^4dx =∫& lt; 0,π/2 & gt; (sinx)^4dx+∫& lt; π/2,π& gt; (sinx)^4dx
For the latter, let x = π-u, and we get
I2 =∫& lt; π/2,π& gt; (sinx)^4dx =∫& lt; π/2,0 & gt(sinu)^4(-du)=∫& lt; 0,π/2 & gt; (sinu)^4du
Definite integral has nothing to do with integral variable. If u is replaced by x, I2 = ∫.
Then I = ∫ < 0, π >; (sinx)^4dx = 2∫& lt; 0,π/2 & gt; (sinx)^4dx