What is the integral from 1 to 2 of (3^{2ln(x)})/x ?

The integral from 1 to 2 of (3^{2ln(x)})/x is (9^{ln(2)}-1)/(2ln(3))

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integral from 1 to 2 of (3^{2ln(x)})/x

\int_{1}^{2} \frac{3 ^{2 l n (x)}}{x} d x

\frac{9 ^{l n (2)} - 1}{2 ln ( 3 )}

Decimal

1.63204 \dots

Solution steps

\int_{1}^{2} \frac{3 ^{2 l n (x)}}{x} d x

3^{2 l n (x)} = 9^{l n (x)}

= \int_{1}^{2} \frac{9 ^{l n (x)}}{x} d x

Apply u-substitution

: \int_{0}^{l n (2)} 9^{u} d u

= \int_{0}^{l n (2)} 9^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= [\frac{9 ^{u}}{ln ( 9 )}]_{0}^{l n (2)}

Simplify

ln (9) : 2 ln (3)

= [\frac{9 ^{u}}{2 ln ( 3 )}]_{0}^{l n (2)}

Compute the boundaries:

\frac{9 ^{l n (2)} - 1}{2 ln ( 3 )}

= \frac{9 ^{l n (2)} - 1}{2 ln ( 3 )}

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What is the integral from 1 to 2 of (3^{2ln(x)})/x ?
The integral from 1 to 2 of (3^{2ln(x)})/x is (9^{ln(2)}-1)/(2ln(3))