Evaluate the integral \( \int \frac{1}{2x} \, dx \).

Evaluate the integral \( \int \sec x \, dx \) using strategic multiplication.

Evaluate the integral $\int \frac{1}{\sqrt{x^2-25}}dx$, where $x>5$, using trigonometric substitution.

Evaluate the integral ${\int }_1^4(1+\ln x)x^{x}dx$.

Evaluate the definite integral ${\int }_1^e\frac{dx}{x\sqrt{{\ln }^{2}x+9}}$. Express your answer in terms of logarithms.