Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

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. 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:

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

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