Tazmin Horton

2021-11-06

Evaluate the following integrals using integration by parts.

${\int}_{1}^{e}\mathrm{ln}2xdx$

Ayesha Gomez

Skilled2021-11-07Added 104 answers

Step 1

Given:$I={\int}_{1}^{e}\mathrm{ln}2xdx$

for evaluating given integral, we use integral by parts

according to integral by parts

$\int f\left(x\right){g}^{\prime}\left(x\right)dx=f\left(x\right)\int {g}^{\prime}\left(x\right)dx-\int [{f}^{\prime}\left(x\right)\int {g}^{\prime}\left(x\right)dx]dx$ ...(1)

here,

$f\left(x\right)=\mathrm{ln}\left(2x\right),g\left(x\right)=1$

Step 2

so, by using equation(1)

$\int \mathrm{ln}\left(2x\right)dx=\mathrm{ln}\left(2x\right)\int dx-\int [\frac{d}{dx}\left(\mathrm{ln}2x\right)\int fx]dx\text{}\text{}\text{}(\because \frac{d}{dx}\left(\mathrm{ln}x\right)=\frac{1}{x})$

$=\mathrm{ln}\left(2x\right)\left[x\right]-\int \frac{1}{2x}\left(2\right)xdx$

$=x\mathrm{ln}\left(2x\right)-\int dx$

$=x\mathrm{ln}\left(2x\right)-x+c$

hence, given integral is equal to$(x\mathrm{ln}\left(2x\right)-x+c)$ .

Given:

for evaluating given integral, we use integral by parts

according to integral by parts

here,

Step 2

so, by using equation(1)

hence, given integral is equal to

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