MATHEMATICS II - Deneme Sınavı - 16

18=q²+2

4=q

Producer surplus is defined as:

we clearly see that

Since the graph of passes through the point we get

2=F(0)=0³+0²-0+c ⇒ c=2.

Hence f(x)=x³+x²-x+2

we substitute and in the integral we have

We need to use the method of integration by parts to find the first integral above. Let u=y and dv=e^ydy, then we get du=dy and v=e^y. Hence it is obvious that

Therefore, we find

We need again method of integration by parts to evaluate ∫x²e^xdx. Let u=x² and dv=e^xdx which means du=2xdx and v=e^x. Hence 

and similarly we have

Thus, we obtain 

and finally we get

and

Hence we get

If we turn to the solution of the question 6, we clearly see that  

Hence we obtain (remembering w²=x and so w³=x√x )

Therefore, we obtain

where c=c₁+c₂+c₃.  After these calculations we obtain the integral as