Question

Evaluate the integral
7x/ x squared +2 [1,5]

Answer 1

\[\int\limits_{1}^{5} 7x/ x^2+2 dx\]

Answer 2

make the substitution
\(x^2+2=u\)

Answer 3

then
\[2xdx=du\implies x\,dx={du\over2}\]
when \(x=1,u=3\)
when \(x=5,u=27\)

Answer 4

Get the integral of the function and solve for x at 5 and 2

Answer 5

i mean 5 and 1

Answer 6

solve the integral in u,
replace u with \(x^2+2\) and apply the limits.

Answer 7

\[
\int_3^{27}{7\over u}{du\over2}={7\over2}[\ln|u|]_3^{27}={7\over2}[\ln(27-\ln(3)]\\
\quad={7\over2}[3\ln(3)-\ln(3)]=\mathbf{7\ln(3)}
\]

Answer 8

I find this way easier