Question

Find the area ^ln(x)⁄x from x = x = ae^b to a.

(a) ^b^2⁄2

(b) ^b⁄2

(c) b

(d) 1

The question was asked in my homework.

Question is taken from Improper Integrals topic in portion Integral Calculus of Engineering Mathematics

Answer 1

Right choice is (a) ^b^2⁄2

To explain: Let, F(x)=\(\int \frac{ln⁡(x)}{x} dx\)

Let, z=ln⁡(x)=>dz=dx/x

=F(x)=∫ zdz=\(\frac{z^2}{2}=\frac{ln^2⁡(x)}{2}\)

Area inside curve from 4a to a is,

\(F(ae^b)-F(a)=\frac{ln^2⁡(ae^b )}{2}-\frac{ln^2⁡(a)}{2}=\frac{ln^2⁡(\frac{ae^b}{a})}{2}=\frac{ln^2⁡(e^b)}{2}=\frac{b}{2}\)