Question

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

Answer 1

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