Calculus
# Integration Techniques

Suppose that $f$ is a smooth function (meaning that all orders of derivatives exist and are continuous) such that $f(0)=1$, $f(2)=3$ and $f' (2) = 5$, then evaluate

$\large \int_{0}^{1} x f'' (2x)\, dx .$

$\int_3^6 \left ( \sqrt{x + \sqrt{12x-36}} + \sqrt{x - \sqrt{12x-36}} \ \right ) \ dx$

If the definite integral above can be expressed as $a \sqrt b$ where $a,b$ are positive integers with $b$ square free, then what is the value of $ab$?

Evaluate the integral $\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^2+2x+4} \, dx.$

Round your answer to three decimal places.

Evaluate:

$\large \displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cot(x)\sec^2(x)}{\cot(x) + 1}\, dx$