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

The integral from 1 to e of (4ln(x))/x is 2

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integral from 1 to e of (4ln(x))/x

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

2

Solution steps

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

Apply u-substitution

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

= 4 \cdot \int_{0}^{1} u d u

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{0}^{1}

= 4 [\frac{u ^{2}}{2}]_{0}^{1}

Compute the boundaries:

\frac{1}{2}

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

Simplify

= 2

\int_{0}^{\infty} t d t

\int_{0}^{1} (5 x^{3} - \frac{1}{2}) d x

\int_{1}^{6} t^{2} - 4 t - 12 d t

\int_{- 7}^{8} \frac{1}{2} x d x

\int_{0}^{6} - \frac{5}{12} x^{2} - 5 x d x

What is the integral from 1 to e of (4ln(x))/x ?
The integral from 1 to e of (4ln(x))/x is 2