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

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

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

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

3

Solution steps

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

Apply Integration By Parts

: [x ln (x^{3}) - \int 3 d x]_{1}^{e}

= [x ln (x^{3}) - \int 3 d x]_{1}^{e}

\int 3 d x = 3 x

= [x ln (x^{3}) - 3 x]_{1}^{e}

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= [3 x ln (x) - 3 x]_{1}^{e}

Compute the boundaries:

3

= 3

\int_{- 4}^{4} - ∣ x ∣ + 4 d x

\int_{3}^{5} (2 x + 9) d x

\int_{8}^{11} 2 x + 3 d x

\int_{0}^{5} \frac{2}{3 x + 2} d x

\int_{1}^{2} \frac{1}{4} x d x

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