A Teaching-and-Academic Awards winner, UPenn PhD top graduate

Ariel B.

Elie,

In the integral e^x sin^2(e^x)cos(e^x)dx

you can first replace e^xdx as d(e^x) and use e^x as a new variable t. Then the integral becomes

Int [sin^2(t) cos(t)]dt . Then you recognize cos(t)dt as -d(sint) and use sint=u as new variable , after which the integral becomes

-Int(sin^2u du) which can be solved by expressing sin^2(u) through cos(2u) ; that integral is a table integral. After you find it, go back from u to t to x

to express your result as a function of x. (And don't miss the - sign

Hope that would help)



Good luck

Dr.Boroda

Hi Ellie,

It is not quite clear what was the question they ask you.

However, I guess it might be about the difference in behavior of two functions near x=0

The function arcsin(3x) has a limit zero when x⇒0 in such a way that /x the ratio arsin(3x)/x has a final limit at x=0 (=3) [one cam use the L'opital Rule: lim [u(x)/v(x)] = lim [u'(x)/v'(x)] when x⇒0 to get it]

The function arcos(x) has final limit (Pi/2) at x=0 ; therefore arccos(x)/x would not have a limit (it goes to - ∞ when x⇒-0 and goes to + ∞ when x⇒+0)

The difference in heneral behavior between two functions is due to the fact that arsin(x) is an odf function of x while arccos(x) is an even function of x: arcsin(-x)=-arcsin(x) but arccos(-x) = + arccos(x)

Hope that'd help

Best,

Dr.Boroda

Hi Oranuo,

When 2 matrices say, a_ik and b_ik are equal it means that a₁₁=b₁₁ , a₁₂=b_{12 etc. In the given matrices only 2 elements contain x or y (and thus allow us to write an equations to determine x and y):}

_{2=x+3. and 2=3y-4}

_{ from which one can find x and y}

_{As to the other two of the 4 elements of the matrices, they present obvious statements: 1=1 and -3=-3}

_{Hope this would help}

_Best,

_Dr.Boroda