Solved: "Evaluate the following integrals. Include absolute values only..."

Kye

Answered question

2021-10-10

Evaluate the following integrals. Include absolute values only when needed.

\int_{1}^{2 e} \frac{3^{\ln x}}{x} d x

Bella

Skilled2021-10-11Added 81 answers

Here given integration as

I = \int_{1}^{2 e} \frac{3^{\ln x}}{x} d x

consider

\ln x = t

\Rightarrow \frac{1}{x} d x = d t

Also

x \to 1, t \to 0

x \to 2 e, t \to \ln (2 e)

Thus integration becomes

I = \int_{0}^{\ln (2 e)} 3^{t} d t

Now

\int a^{t} d t = \frac{a^{t}}{\ln a} + C

Thus

I = {[\frac{3^{t}}{\ln 3}]}_{0}^{\ln (2 e)}

\Rightarrow I = \frac{1}{\ln 3} [3^{\ln 2 e} - 1]

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