If for n 1, P _ n = ₁^ e ( x ^ n ) dx , then P ₁₀-90 P ₈ is equal to:

Step-by-step solution

P _ n = ₁^e( x)^n d x put x=t then x=e^t and d x=e^t d t Also, when x=1 , then t= 1=0 and when x=e , then t= _e e=1 gathered P _ n = ₀^1 t^n e^t d t P ₁₀= ₀^1 t¹⁰ e^t d t and P ₈= ₀^1 t^8 e^t d t Now, P ₁₀-90 P ₈= ₀^1 t¹⁰ e^t d t-90 ₀^1 t^8 e^t d t P ₁₀-90 P ₈ P ₁₀-90 P ₈ = [t¹⁰ e^t ]₀^1-10 ₀^1 t^9 e^t d t-90 ₀^1 t^8 e^t d t P ₁₀-90 P ₈= e-10 [t^9 ₀^1 e^t d t- ₀^1 d d t (t^9 ) e^t d t ]-90 ₀^1 t^8 e^t d t gathered gathered P ₁₀-90 P ₈=e-10 [e-9 ₀^1 t^8 e^t d t ]-90 ₀^1 t^8 e^t d t P ₁₀-90 P ₈=e-10 e+90 t^8 e^t d t-