Calculate the area under the graph of f(x)=-ln(x) above the x-axis on the interval (0,1] ∞ 2 1 0

Question

Calculate the area under the graph of
`f(x)=-ln(x)` above the x-axis on the interval (0,1]

∞
2
1
0

Answer 1

1

Explanation:

∫ -ln x dx

To integrate this, we must integrate by parts:

∫ u dv = uv − ∫ v du

If we say u = ln x and dv = -dx,

then du = dx / x and v = -x.

∫ -ln x dx = -x ln x − ∫ -dx

∫ -ln x dx = -x ln x + x + C

The upper bound of the integral is x = 1.

The lower bound of the integral is x = t.

(-1 ln 1 + 1 + C) − (-t ln t + t + C)

1 + t ln t − t

If we take the limit as t approaches 0:

lim(t→0) (1 + t ln t − t)

1 + lim(t→0) (t ln t)

1 + lim(t→0) (ln t / (1/t))

Using L'Hopital's rule:

1 + lim(t→0) ((1/t) / (-1/t²))

1 + lim(t→0) (-t)

1