Calculus help!?

Question

What is the integral of log base 5 of x?

Answer 1

log base 5 of x = lnx/ln5

Using I = integral

1/ln5 is a constant so I(lnx/ln5) = (1/ln5)I(lnx)

Now I(lnx) = xlnx - x + C

So answer is (xlnx-x+C)/ln5

Answer 2

So you want to solve

Integral (log [base 5](x)dx

As others have pointed out, you want to use the Change of Base formula to convert this. Recall that the change of base formula goes as follows:

log[base c](a) = log[base b](a) / log[base b])(c)

This implies we can choose a new base from an old base. In our case, what we want to choose is base e. This would give us

log[base 5](x) = log[base e](x) / log[base e](5)

and log[base e] is defined to be ln, so

log[base 5](x) = ln(x) / ln(5) = [1/ln(5)] ln(x)

Therefore,
Integral (log[base 5](x))dx = Integral ( [1/ln(5)] ln(x) )dx

When solving any integral, it's best to pull out constants.

[1/ln(5)] * Integral (ln(x))dx

And now we solve the integral of ln(x) using integration by parts.

Let u = ln(x). dv = dx
du = (1/x) dx. v = x

The result of integrating by parts would give us uv - Integral (v du)

[1/ln(5)] { xln(x) - Integral ( (1/x)x )dx }

Note that (1/x) times x is equal to 1.

[1/ln(5)] {xln(x) - Integral (1) dx }

Giving us the solution

[1/ln(5)] {xln(x) - x} + C

Expanding this out,

x ln(x)/ln(5) - x/ln(5) + C

Answer 3

hmm... try integration by parts. Consider the integrand to be 1 * log_5 (x) and then set...

u = log_5 (x)
dv = 1 dx

And see where that takes you.

Or, as suggested above, use the change of base formula to make it ln (x) / ln_e (x). Then pull out 1 / ln_e (x) as a constant in front of the integral sign and you are left with the integral of ln (x). If you use the integration by parts trick I mentioned above (by looking at it as 1 * ln (x) ) then it will work out.

Answer 4

the answer is 1/(xln5)

you take the antiderivative of x (which is 1) and put that over the natural log of the base of the log (ln5) times x.

I hope that makes sense.

Answer 5

y= f(x) = log base 5 of x
This is the same as saying:
5^y =x
So ln 5^y = ln x
y ln 5 = ln x
y=(ln x)/(ln 5)
integ y = f(x) = integ (ln x)/(ln 5)
=(1/ln 5) integ ln x
= (1/ln 5)91/x) + C

Answer 6

∫lnx/ln5 dx
=(1/ln5)(xlnx - ∫dx) (integration by parts)
=(1/ln5)(xlnx - x) + c

Answer 7

plesae visit this link for quick solution:
http://calc101.com/parts_2.html

Answer 8

http://www.public.iastate.edu/~aaallen/Mat165-E1Fa05/Mat165Ch7Formulas.pdf