integral of ln(x)(sqrt(1-(ln(x))^2))/x

⚠Ⓘ ln(x)(sqrt(1-(ln(x))^2))/x is read as ln(x)*(sqrt(1-(ln(x))^2))/x: use (...)*(...) for multiplication, use (...)^(...) for a power

1) Let's integrate:

\int(\frac{\ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x})dx

For the math expression:

\frac{\ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x}

, we know the derivative formula:

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, so we try differentiating with ^(m+1)

2) Let's differentiate:

\frac{d}{d x} ({(1 - {\ln (x)}^{2})}^{(\frac{1}{2} + 1)})

\frac{d}{d x} ({(1 - {\ln (x)}^{2})}^{(\frac{1 + 2}{2})})

\frac{d}{d x} ({(1 - {\ln (x)}^{2})}^{\frac{3}{2}})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

with f =

1 - {\ln (x)}^{2}

\frac{3}{2} \cdot \frac{d}{d x} (1 - {\ln (x)}^{2}) \cdot {(1 - {\ln (x)}^{2})}^{(\frac{3}{2} - 1)}

ℹ

\frac{3}{2} \cdot \frac{d}{d x} (1 - {\ln (x)}^{2}) \cdot {(1 - {\ln (x)}^{2})}^{(\frac{3}{2} - 1)}

\frac{3 \cdot \frac{d}{d x} (1 - {\ln (x)}^{2}) \cdot {(1 - {\ln (x)}^{2})}^{(\frac{3}{2} - 1)}}{2}

‖

\frac{d}{d x} (1 - {\ln (x)}^{2})

0 + \frac{d}{d x} (- {\ln (x)}^{2})

‖

\frac{3}{2} - 1

\frac{3 - 1 \cdot 2}{2}

\frac{3 \cdot (0 + \frac{d}{d x} (- {\ln (x)}^{2})) \cdot {(1 - {\ln (x)}^{2})}^{(\frac{3 - 1 \cdot 2}{2})}}{2}

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

(where f =

\ln (x)

)

\frac{3 \cdot (- 2 \cdot \frac{d}{d x} (\ln (x)) \cdot \ln (x)) \cdot {(1 - {\ln (x)}^{2})}^{(\frac{3 - 2}{2})}}{2}

ℹ

3 \cdot (-2)

-6

‖

\frac{d}{d x} (\ln (x))

\frac{1}{x}

‖

3 - 2

1

\frac{-6 \cdot \frac{1}{x} \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{2}

\frac{\frac{-6 \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x}}{2}

\frac{- \frac{6 \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x}}{2}

- \frac{\frac{6 \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x}}{2}

- \frac{6 \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x} \cdot \frac{1}{2}

- \frac{6 \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x \cdot 2}

- \frac{3 \cdot \ln (x) \cdot \sqrt{1 - {\ln (x)}^{2}}}{x}

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

\frac{1}{-3} \cdot {(1 - {\ln (x)}^{2})}^{\frac{3}{2}}

- \frac{1}{3} \cdot {(1 - {\ln (x)}^{2})}^{\frac{3}{2}}

- \frac{{(1 - {\ln (x)}^{2})}^{\frac{3}{2}}}{3}

Indefinite integrals are defined up to an additive constant C, so this yields:

- \frac{{(1 - {\ln (x)}^{2})}^{\frac{3}{2}}}{3} + C

▶Antiderivative:

- \frac{{(- {\ln (x)}^{2} + 1)}^{\frac{3}{2}}}{3} + C

, where C is a constant