What is the integral from 1 to 3 of (4(ln(x))^3)/x ?

The integral from 1 to 3 of (4(ln(x))^3)/x is ln^4(3)

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integral from 1 to 3 of (4(ln(x))^3)/x

\int_{1}^{3} \frac{4 ( ln ( x ) ) ^{3}}{x} d x

ln^{4} (3)

Decimal

1.45672 \dots

Solution steps

\int_{1}^{3} \frac{4 ( ln ( x ) ) ^{3}}{x} d x

Take the constant out:

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

= 4 \cdot \int_{1}^{3} \frac{( ln ( x ) ) ^{3}}{x} d x

Apply u-substitution

= 4 \cdot \int_{0}^{l n (3)} u^{3} d u

Apply the Power Rule

= 4 [\frac{u ^{4}}{4}]_{0}^{l n (3)}

Compute the boundaries:

\frac{ln ^{4} ( 3 )}{4}

= 4 \cdot \frac{ln ^{4} ( 3 )}{4}

Simplify

= ln^{4} (3)

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