Expert Answer to "Evaluate the following integrals using integration by parts...."

Tazmin Horton

Answered question

2021-11-06

Evaluate the following integrals using integration by parts.

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

Ayesha Gomez

Skilled2021-11-07Added 104 answers

Step 1
Given:

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

for evaluating given integral, we use integral by parts
according to integral by parts

\int f (x) g^{'} (x) d x = f (x) \int g^{'} (x) d x - \int [f^{'} (x) \int g^{'} (x) d x] d x

...(1)
here,

f (x) = \ln (2 x), g (x) = 1

Step 2
so, by using equation(1)

\int \ln (2 x) d x = \ln (2 x) \int d x - \int [\frac{d}{d x} (\ln 2 x) \int f x] d x (∵ \frac{d}{d x} (\ln x) = \frac{1}{x})

= \ln (2 x) [x] - \int \frac{1}{2 x} (2) x d x

= x \ln (2 x) - \int d x

= x \ln (2 x) - x + c

hence, given integral is equal to

(x \ln (2 x) - x + c)

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?