Question

how to integrate log10x

Answer 1

I=∫log|10x|dxI=∫log⁡|10x|dx

I=110∫10log|10x|dxI=110∫10log⁡|10x|dx

Substitute:

10x=t10x=t

⟹10dx=dt⟹10dx=dt

⟹I=110∫log|t|dt⟹I=110∫log⁡|t|dt

Apply by part integration formula.

⟹I=110(log|t|∫1dt−∫(ddt(log|t|)⋅∫1dt)dt)⟹I=110(log⁡|t|∫1dt−∫(ddt(log⁡|t|)⋅∫1dt)dt)

⟹I=110(tlog|t|−∫(1t⋅t)dt)⟹I=110(tlog⁡|t|−∫(1t⋅t)dt)

⟹I=110(tlog|t|−∫1dt)⟹I=110(tlog⁡|t|−∫1dt)

⟹I=110(tlog|t|−t)+c⟹I=110(tlog⁡|t|−t)+c

Converting back to original variable xx

⟹I=110(10xlog|10x|−10x)+C⟹I=110(10xlog⁡|10x|−10x)+C

⟹I=xlog|10x|−x+C⟹I=xlog⁡|10x|−x+C

⌣¨


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