What is the integral from 1 to e of (8(ln(x))^3)/x ?

The integral from 1 to e of (8(ln(x))^3)/x is 2

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integral from 1 to e of (8(ln(x))^3)/x

\int_{1}^{e} \frac{8 ( ln ( x ) ) ^{3}}{x} d x

2

Solution steps

\int_{1}^{e} \frac{8 ( ln ( x ) ) ^{3}}{x} d x

Take the constant out:

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

= 8 \cdot \int_{1}^{e} \frac{( ln ( x ) ) ^{3}}{x} d x

Apply u-substitution

: \int_{0}^{1} u^{3} d u

= 8 \cdot \int_{0}^{1} u^{3} d u

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{0}^{1}

= 8 [\frac{u ^{4}}{4}]_{0}^{1}

Compute the boundaries:

\frac{1}{4}

= 8 \cdot \frac{1}{4}

Simplify

= 2

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

\int_{- 1}^{2} \frac{1}{x} d x

\int_{- 1}^{0} e^{x} d x

\int_{\frac{1}{4}}^{2} \frac{ln ( x )}{x} d x

\int_{3}^{8} ln (1 + x) d x

What is the integral from 1 to e of (8(ln(x))^3)/x ?
The integral from 1 to e of (8(ln(x))^3)/x is 2