What is the integral of 5X ln(3+X^2) dx?

Question

I know you can use synthetic division but can anyone show how to do this using long division only? is it possible

Answer 1

why not use substitution?

substitute

z = 3 + x²

=>

dz = 2x dx

∫ 5∙x∙ln(3+x²) dx

=  Ã¢ÂˆÂ« 5 ∙ x ∙ ln(z) ∙(1/(2∙x)) dz

= (5/2) ∙ ∫ ln(z) dz

integrate by parts with

u'=1 → u=z

v=ln(z) → v'=1/z

= (5/2) ∙ ( z∙(ln(z)  - ∫ z∙(1/z) dz )

= (5/2) ∙ ( z∙(ln(z)  - z) + c

= (5/2) ∙ z ∙ ( (ln(z)  - 1) + c

= (5/2) ∙ (3+x²) ∙ (ln(3+x²) - 1) + c