(e^ x _e 6 ) e^x d x= (x)+c then (x)=

Step-by-step solution

The integral is given by I = (e^ x _ e 6 ) e^x dx . Using the property e^ x _ e a = (e^ _ e a )^x = a^x , we have e^ x _ e 6 = 6^x . Substituting this into the integral, I = 6^x e^x dx = (6e)^x dx . Using the standard integral formula a^x dx = a^x _ e a + c , where a = 6e , we get: I = (6e)^x _ e (6e) + c . Using the property _ e (ab) = _ e a + _ e b , we have _ e (6e) = _ e 6 + _ e e = _ e 6 + 1 . Therefore, (x) = (6e)^x 1 + _ e 6 . Answer: (6 e)^x 1+ _e 6