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

The integral of ((ln(x))^3)/(2x) is 1/8 ln^4(x)+C

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

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

\frac{1}{8} ln^{4} (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int u^{3} d u

= \frac{1}{2} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{1}{2} \cdot \frac{u ^{4}}{4}

Substitute back

u = ln (x)

= \frac{1}{2} \cdot \frac{ln ^{4} ( x )}{4}

Simplify

\frac{1}{2} \cdot \frac{ln ^{4} ( x )}{4} : \frac{1}{8} ln^{4} (x)

= \frac{1}{8} ln^{4} (x)

Add a constant to the solution

= \frac{1}{8} ln^{4} (x) + C

\frac{\partial}{\partial x} (cos (x) sin (x))

(\frac{1}{2}, 7), (3, - \frac{3}{2})

f (t) = \frac{( t ^{2} - 4 t )}{t + 3}

\int \frac{2 x + 3}{x ^{2} + 3 x + 4} d x

x \to 0 + lim (x^{n})

What is the integral of ((ln(x))^3)/(2x) ?
The integral of ((ln(x))^3)/(2x) is 1/8 ln^4(x)+C