Step one: State the integral to evaluate
\(\int_{1}^{2}e^{2 \ln x}dx\)
Step two: Apply the logarithm power rule
\(2 \ln x=\ln (x^2)\)
Rewrite
\(e^{\ln(x^2)}\)
Step three: Cancel inverse functions
\(e^{ln(x^2)}=x^2\)
Sub into integral:
\(I=\int^{2}_{1} x^2 dx\)
Evaluate
\(\int x^2dx=\frac{x^3}{3}+C\)
Apply the fundamental theorem of calculus, calculate the upper minus lower bound antiderivative
\(\frac{2^3}{3} = \frac{8}{3}\), \(\frac{1^3}{3} = \frac{1}{3}\),
\(\boxed{\frac{8}{3}-\frac{1}{3} = \frac{7}{3}}\)
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