Get the answer to "Use the basic integration rules to find or..."

Josalynn

Answered question

2021-10-26

Use the basic integration rules to find or evaluate the integral.

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

yunitsiL

Skilled2021-10-27Added 108 answers

Step 1
Given:

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

for evaluating given integral, we substitute

\ln (2 x) = t

...(1)
now, differentiating equation(1) with respect to x

\frac{d}{d x} (\ln (2 x)) = \frac{d t}{d x} (∵ \frac{d}{d x} (\ln x) = \frac{1}{x})

\frac{1}{2 x} (2) = \frac{d t}{d x}

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

Step 2
now, replacing

\frac{d x}{x}

with dt,

\ln (2 x)

with t and change limits also in given integral
so,

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

= \int_{\ln 2}^{1 + \ln 2} t d t

= {(\frac{t^{2}}{2})}_{\ln 2}^{1 + \ln 2}

= \frac{1}{2} [{(1 + \ln 2)}^{2} - {(\ln 2)}^{2}] (∵ {(a + b)}^{2} = a^{2} + 2 a b + b^{2})

= \frac{1}{2} [1 + 2 \ln 2 + {(\ln 2)}^{2} - {(\ln 2)}^{2}]

= \frac{1}{2} [1 + 2 \ln 2]

hence, given integral is equal to

\frac{1}{2} (1 + 2 \ln 2)

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