How do you Integrate ln(f(x)) or x*sqrt(f(x))? I am stuck, any sort of help will be appreciated, thanks! =)?

Question

I am specifically trying to integrate:

ln(x^2 +4)
and
x*sqrt(3-x)

but I want to do the work myself, so IF you want to answer my specific question, put it in this form:

[how to integrate ln(f(x) and x*sqrt(f(x))]
[huge space so i can know the following is answer]
[answer to the 2 specific integrals]

thanks alot! ^_^

Answer 1

For the first one, try integration by parts, to reduce
it to integrating a rational function.
For the second, let u = 3-x, dx = -du.
You can get the answers on integrals.wolfram.com
In general, you cannot always integrate
ln(f(x)) or x*sqrt(f(x)) in terms of elementary
functions.
For example, ln(sin(x)) and ln(cos(x)) do not
have elementary antiderivatives.

What about x*sqrt(f(x))? In general, if f(x)
is a polynomial of degree 3 or greater, the
integral of x*sqrt(f(x)) is not elementary.
Just for fun, try to integrate x*sqrt(x^3+1)! Its
solution involves both the first and second kinds
of elliptic integral!

Answer 2

take u=ln(x^2+4) and dv as 1 and try

for the second one put (3-x)=t dx=-dt
x=3-t
so the integral will be
=(3-t)*t^1/2 (-dt)

Answer 3

just use the chain rule:
y = f(x)
y' = f(x)*f'(x)

so for ur question:
y = ln(x^2 + 4)
y' = (1/(x^2 + 4))*(2x)


and

for y=x*sqrt(x-3) use product rule
y = g(x)h(x)
y' = g'(x)h(x) + h'(x)g(x)

for ur question: g(x) = x and h(x) =sqrt(x-3)
y = x*sqrt(x-3)
y'= (1)(sqrt(x-3) + (1/2)((sqrt(x-3)^(-1/2))(1)
and simplify

Answer 4

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