What is the integral from 10 to 19 of ln(x^{15}) ?

The integral from 10 to 19 of ln(x^{15}) is 15(19ln(19)-9-10ln(10))

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integral from 10 to 19 of ln(x^{15})

\int_{10}^{19} ln (x^{15}) d x

15 (19 ln (19) - 9 - 10 ln (10))

Decimal

358.77734 \dots

Solution steps

\int_{10}^{19} ln (x^{15}) d x

Apply log rule

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

assuming

x \geq 0

= \int_{10}^{19} 15 ln (x) d x

Take the constant out:

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

= 15 \cdot \int_{10}^{19} ln (x) d x

Apply Integration By Parts

: [x ln (x) - \int 1 d x]_{10}^{19}

= 15 [x ln (x) - \int 1 d x]_{10}^{19}

\int 1 d x = x

= 15 [x ln (x) - x]_{10}^{19}

Compute the boundaries:

19 ln (19) - 9 - 10 ln (10)

= 15 (19 ln (19) - 9 - 10 ln (10))

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What is the integral from 10 to 19 of ln(x^{15}) ?
The integral from 10 to 19 of ln(x^{15}) is 15(19ln(19)-9-10ln(10))