Question Number 101650 by I want to learn more last updated on 03/Jul/20

Commented bymr W last updated on 03/Jul/20

f(x)=a^x for a∈R^+

Answered by mathmax by abdo last updated on 03/Jul/20

I =∫_(−1) ^1 (dx/(1+f(x))) cha7gement x =−t give I =−∫_(−1) ^1 ((−dt)/(1+f(−t))) =∫_(−1) ^1 (dt/(1+(1/(f(t))))) =∫_(−1) ^1 ((f(t))/(1+f(t))) dt ⇒2I =∫_(−1) ^1 (dx/(1+f(x))) +∫_(−1) ^1 ((f(x))/(1+f(x)))dx =∫_(−1) ^1 dx =2 ⇒ I =1

Commented byI want to learn more last updated on 04/Jul/20

Thanks sir

Commented bymathmax by abdo last updated on 04/Jul/20

you are welcome

Answered by MAB last updated on 03/Jul/20

∫_(−1) ^0 (1/(1+f(x)))dx=^(u=−x) ∫_0 ^1 (1/(1+f(−u)))du =∫_0 ^1 (1/(1+(1/(f(u)))))du (f(−u)=(1/(f(u)))) =∫_0 ^1 ((f(u))/(1+f(u)))du thus ∫_(−1) ^1 (1/(1+f(x)))dx=∫_0 ^1 (1/(1+f(x)))+((f(x))/(1+f(x)))dx =∫_0 ^1 1dx =1

Commented byMAB last updated on 05/Jul/20

you are welcome

Commented bymr W last updated on 03/Jul/20

very nice sir! ∫_(−1) ^1 (dx/(1+a^x ))=1 for any a∈R^+

Commented byI want to learn more last updated on 04/Jul/20

Thanks sir.