What is the integral of (ln(x^{3/2}))/x ?

The integral of (ln(x^{3/2}))/x is 3/4 ln^2(x)+C

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integral of (ln(x^{3/2}))/x

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

\frac{3}{4} ln^{2} (x) + C

Solution steps

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

Simplify

\frac{ln ( x ^{\frac{3}{2}} )}{x} : \frac{3 ln ( x )}{2 x}

= \int \frac{3 ln ( x )}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{3}{2} \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

= \frac{3}{2} \cdot \int u d u

Apply the Power Rule

= \frac{3}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

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

Simplify

\frac{3}{2} \cdot \frac{ln ^{2} ( x )}{2} : \frac{3}{4} ln^{2} (x)

= \frac{3}{4} ln^{2} (x)

Add a constant to the solution

= \frac{3}{4} ln^{2} (x) + C

\int_{0}^{t} 3 e^{s} sin (t - s) d s

\int \frac{1}{2 x + 3} d x

\int (cos (x) sin (x)) d x

x \to 0 lim (\frac{2 x}{∣ x ∣})

\frac{d y}{d x} = (3 x y)

What is the integral of (ln(x^{3/2}))/x ?
The integral of (ln(x^{3/2}))/x is 3/4 ln^2(x)+C