Integration Techniques

Integration Techniques: Level 3 Challenges


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

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

The value of the integral \[\int_0^{\pi/2}\dfrac{\text dx}{2+\cos x}\] can be expressed in the form \(\dfrac{\pi^A\sqrt{B}}{C}\) where \(A\), \(B\), \(C\) are positive integers and \(B\) is not divisible by the square of any prime. Find the value of \(A+B+C.\)

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

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.


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


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