What is the integral from 1 to 4 of ln(sqrt(x)) ?

The integral from 1 to 4 of ln(sqrt(x)) is 4ln(2)-3/2

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integral from 1 to 4 of ln(sqrt(x))

\int_{1}^{4} ln (x) d x

4 ln (2) - \frac{3}{2}

Decimal

1.27258 \dots

Solution steps

\int_{1}^{4} ln (x) d x

Apply Integration By Parts

: [x ln (x) - \int \frac{1}{2} d x]_{1}^{4}

= [x ln (x) - \int \frac{1}{2} d x]_{1}^{4}

\int \frac{1}{2} d x = \frac{1}{2} x

= [x ln (x) - \frac{1}{2} x]_{1}^{4}

Simplify

[x ln (x) - \frac{1}{2} x]_{1}^{4} : [\frac{1}{2} (x ln (x) - x)]_{1}^{4}

= [\frac{1}{2} (x ln (x) - x)]_{1}^{4}

Compute the boundaries:

4 ln (2) - \frac{3}{2}

= 4 ln (2) - \frac{3}{2}

\frac{d}{d x} (e^{2 x^{5}})

\int \frac{1}{4 + 5 cos ( x )} d x

\int cos (x) e^{a x} d x

t (ln (2 t))^{2}

3 x (a^{2} - x^{2})

What is the integral from 1 to 4 of ln(sqrt(x)) ?
The integral from 1 to 4 of ln(sqrt(x)) is 4ln(2)-3/2