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

The integral of (3x)/(ln(x)) is 3\Ei(2ln(x))+C

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

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

3 Ei (2 ln (x)) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{e ^{2 u}}{u} d u

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

Apply u-substitution

: \int \frac{e ^{v}}{v} d v

= 3 \cdot \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= 3 Ei (v)

Substitute back

= 3 Ei (2 ln (x))

Add a constant to the solution

= 3 Ei (2 ln (x)) + C

x e^{2 x}

\int \frac{3}{( 1 + 9 x ^{2} )} d x

\int x (a + b x^{3})^{2} d x

\int \frac{1}{( x ^{4} - 1 ) x} d x

\int \frac{1}{2 x ^{2} - x ^{3}} d x

What is the integral of (3x)/(ln(x)) ?
The integral of (3x)/(ln(x)) is 3\Ei(2ln(x))+C