Write the integral in terms of u and du. Then evaluate. ∫(ln⁡x)2dxx,u=ln⁡x

Expert Answer to "Write the integral in terms of u and..."

Khadija Wells

2021-11-10

Write the integral in terms of u and du. Then evaluate.

\int \frac{{(\ln x)}^{2} d x}{x}, u = \ln x

lamusesamuset

Skilled2021-11-11Added 93 answers

Step 1
the given integral is:

\int \frac{{(\ln x)}^{2} d x}{x}, u = \ln x

we have to write the integral in terms of u and du.
then we have to evaluate the integral.
let the given integral be I.
therefore,

I = \int \frac{{(\ln x)}^{2} d x}{x}

Step 2
let

u = \ln x

therefore,

d (u) = d (\ln x)

d u = \frac{1}{x} d x

now substitute these values in the integral I.
therefore,

I = \int \frac{{(\ln x)}^{2} d x}{x}

= \int u^{2} d u

= \frac{u^{2 + 1}}{2 + 1} + C

= \frac{u^{3}}{3} + C

where C is the constant of integration.
Step 3
now substitute the value of u that is u=lnx in the integral I.
therefore,

I = \frac{u^{3}}{3} + C

= \frac{{(\ln x)}^{3}}{3} + C

therefore the value of the given integral

\int \frac{{(\ln x)}^{2} d x}{x} i s \frac{{(\ln x)}^{3}}{3} + C

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