Integration Techniques

Integration Techniques: Level 3 Challenges


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

01xf(2x)dx. \large \int_{0}^{1} x f'' (2x)\, dx .

The value of the integral 0π/2dx2+cosx\int_0^{\pi/2}\dfrac{\text dx}{2+\cos x} can be expressed in the form πABC\dfrac{\pi^A\sqrt{B}}{C} where AA, BB, CC are positive integers and BB is not divisible by the square of any prime. Find the value of A+B+C.A+B+C.

36(x+12x36+x12x36 ) 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 aba \sqrt b where a,ba,b are positive integers with bb square free, then what is the value of abab?

Evaluate the integral 0lnxx2+2x+4dx.\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^2+2x+4} \, dx.

Round your answer to three decimal places.


π6π3cot(x)sec2(x)cot(x)+1dx\large \displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cot(x)\sec^2(x)}{\cot(x) + 1}\, dx


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