What is the integral from 6 to 10 of ln(x^{13}) ?

The integral from 6 to 10 of ln(x^{13}) is 13(ln(156250000/729)-4)

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integral from 6 to 10 of ln(x^{13})

\int_{6}^{10} ln (x^{13}) d x

13 (ln (\frac{156250000}{729}) - 4)

Decimal

107.57882 \dots

Solution steps

\int_{6}^{10} ln (x^{13}) d x

Apply log rule

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

assuming

x \geq 0

= \int_{6}^{10} 13 ln (x) d x

Take the constant out:

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

= 13 \cdot \int_{6}^{10} ln (x) d x

Apply Integration By Parts

: [x ln (x) - \int 1 d x]_{6}^{10}

= 13 [x ln (x) - \int 1 d x]_{6}^{10}

\int 1 d x = x

= 13 [x ln (x) - x]_{6}^{10}

Compute the boundaries:

10 ln (10) - 4 - 6 ln (6)

= 13 (10 ln (10) - 4 - 6 ln (6))

Simplify

= 13 (ln (\frac{156250000}{729}) - 4)

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What is the integral from 6 to 10 of ln(x^{13}) ?
The integral from 6 to 10 of ln(x^{13}) is 13(ln(156250000/729)-4)