What is the integral from 0 to 4 of (ln(x))/(sqrt(x)) ?

The integral from 0 to 4 of (ln(x))/(sqrt(x)) is 8ln(2)-8

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integral from 0 to 4 of (ln(x))/(sqrt(x))

\int_{0}^{4} \frac{ln ( x )}{x} d x

8 ln (2) - 8

Decimal

- 2.45482 \dots

Solution steps

\int_{0}^{4} \frac{ln ( x )}{x} d x

Apply Integration By Parts

: [2 x ln (x) - \int \frac{2}{x} d x]_{0}^{4}

= [2 x ln (x) - \int \frac{2}{x} d x]_{0}^{4}

\int \frac{2}{x} d x = 4 x

= [2 x ln (x) - 4 x]_{0}^{4}

Compute the boundaries:

8 ln (2) - 8

= 8 ln (2) - 8

sin^{2} (x)

\frac{d}{d t} (3 cos (t))

x \to 0 lim (3 ln (x))

\int sin^{3} (t) cos^{8} (t) d t

\int_{0}^{\frac{π}{3}} sin (3 t) d t

What is the integral from 0 to 4 of (ln(x))/(sqrt(x)) ?
The integral from 0 to 4 of (ln(x))/(sqrt(x)) is 8ln(2)-8