Question

Substitution formula to evlauate the integral. Posted below

Answer 1

\[\int\limits_{0}^{\ln^\sqrt{3}/3 } \frac{ 3 e^3x dx }{ 1 + e6x }\]

Answer 2

Does it HAVE to be substitution?

\(\int\limits \dfrac{3e^{3}x}{1+e^{6}x}\;dx = \int \dfrac{3e^{3}x}{e^{6}}\; d\left(1+e^{6}x\right) = \dfrac{3x}{e^{3}}\cdot \left(1+e^{6}x\right) - \int \left(1+e^{6}x\cdot \dfrac{3}{e^{3}}\right)\;dx\)

Looks to me like the constants are a little annoying. Where else are you struggling?

You could try \(u = 1 + e^{6}x\) instead of "parts" if you wanted