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# Integration Techniques

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.

# 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. What is the value of $$ab$$?

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

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