Integrate ∫ log⁡x^2 dx

Category: QuestionsIntegrate ∫ log⁡x^2 dx
Editor">Editor Staff asked 11 months ago

Integrate ∫ log⁡x^2 dx
(a) log⁡x^2 + x+C
(b) x log⁡x^2 – 2x+C
(c) x log⁡x^2 – 1+C
(d) x log⁡x^2 + x+C
This question was addressed to me in homework.
This intriguing question comes from Integration by Parts topic in chapter Integrals of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

Right answer is (b) x log⁡x^2 – 2x+C
The best I can explain: By using ∫ u.v dx=u∫ v dx-∫ u'(∫ v dx)
∫ log⁡x^2.1 dx=log⁡x^2 ∫ dx-\(\int \frac{1}{x^2}.2x \int dx\)
=x log⁡x^2 – 2∫ 1/x.x dx
=x log⁡x^2 – 2∫ dx
=x log⁡x^2 – 2x+C