How do you find the integral from e^2 to e of #dx / (x * ln (x^8))#?

Question

Samantha Adkison · Accepted Answer

#-1/8ln2, or, -ln2/8.#

By applying the standard Log function rules, we determine that the Integrand is

#1/{(x)(lnx^8)}=1/{x(8lnx)}=1/(8xlnx)#

Hence, #I=1/8int_(e^2) ^e 1/(xlnx)dx=1/8int_(e^2) ^e (1/(lnx))(1/xdx)#

Substitute, #ln x=t rArr 1/xdx=dt#

Also, #x=e^2 rArr t=ln x=ln e^2=2, &, x=e rArr t=1.#

#:. I=1/8 int_2^1 1/t dt=1/8[ln|t|]_2^1=1/8(ln1-ln2)#

#"Therefore, I="-ln2/8.#

Have fun with math!

Emilia Aguilar · Answer

undefined To find the integral ∫(dx / (x * ln(x^8))) from e^2 to e, you can use the substitution method. Let u = ln(x^8), then du = (8/x) dx. Rearranging this, we get dx = (x/8) du. Substituting these into the integral, we have ∫(dx / (x * ln(x^8))) = ∫(1/(8u)) du. Now, this integral becomes ∫(1/(8u)) du = (1/8) * ln|u| + C = (1/8) * ln|ln(x^8)| + C. To evaluate the definite integral from e^2 to e, plug in the upper and lower limits: (1/8) * [ln|ln(e^8)| - ln|ln(e^2)|] = (1/8) * [ln|8| - ln|2|] = (1/8) * (ln(8) - ln(2)) = (1/8) * ln(4) = ln(√2) / 4. Therefore, the integral from e^2 to e of dx / (x * ln (x^8)) equals ln(√2) / 4.