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[tex]\displaystyle\int\frac{(\ln x)^{44}}x\,\mathrm dx[/tex]

[tex]y=\ln x\implies\mathrm dy=\dfrac{\mathrm dx}x[/tex]

[tex]\displaystyle\int y^{44}\,\mathrm dy=\frac{y^{45}}{45}+C[/tex]

[tex]\displaystyle\int\frac{(\ln x)^{44}}x\,\mathrm dx=\frac{(\ln x)^{45}}{45}+C[/tex]

The value of the integral is (ln x)^45/45 + c.

What is an integration?

The integration is the inverse process of differentiation. The integration is the process of finding the anti derivative of a function. Integration is used to add small and discrete data, which cannot be added singularly and representing in a single value.

For the given situation,

Evaluation of integral [(ln x)^44/x]dx

⇒ [tex]\int\limits{\frac{(ln x)^{44}}{x} \, dx[/tex]

Consider [tex]y=ln x[/tex], differentiate with respect to x

⇒ [tex]dy=\frac{1}{x} dx[/tex]

Substitute these values on the integral,

⇒ [tex]\int\limits {y^{44} } \, dy[/tex]

⇒ [tex]\frac{y^{45} }{45} +c[/tex]

⇒ [tex]\frac{(lnx)^{45} }{45} +c[/tex]

Hence we can conclude that the value of the integral is (ln x)^45/45 + C.

Learn more about integrals here

https://brainly.com/question/18125359

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