math calc questions help

Question

1. integral e^x sin^2(e^x)cos(e^x)dx
2. if k(x) = x √(e^x-3), what is k'(x)

i'm not sure how to approach these questions any help is appreciated!

Answer 1

∫ sin²(e^x)cos(e^x)e^xdx

Let u = sin(e^x). Then du = cos(e^x)e^xdx

So, we have ∫u²du = (1/3)u³ + C = (1/3)sin³(e^x) + C

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For the second problem, apply the product rule.

Answer 2

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