What is the integral from 1 to e of ln(x^6) ?

The integral from 1 to e of ln(x^6) is 6

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integral from 1 to e of ln(x^6)

\int_{1}^{e} ln (x^{6}) d x

6

Solution steps

\int_{1}^{e} ln (x^{6}) d x

Apply Integration By Parts

: [x ln (x^{6}) - \int 6 d x]_{1}^{e}

= [x ln (x^{6}) - \int 6 d x]_{1}^{e}

\int 6 d x = 6 x

= [x ln (x^{6}) - 6 x]_{1}^{e}

Apply log rule

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

assuming

x \geq 0

= [6 x ln (x) - 6 x]_{1}^{e}

Compute the boundaries:

6

= 6

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What is the integral from 1 to e of ln(x^6) ?
The integral from 1 to e of ln(x^6) is 6