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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.
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
\[ \large \int_{0}^{1} x f'' (2x)\, dx . \]
by
Sambhrant Sachan
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 \(A+B+C.\)
by
Trevor B.
\[ \int_3^6 \left ( \sqrt{x + \sqrt{12x-36}} + \sqrt{x - \sqrt{12x-36}} \ \right ) \ dx \]
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 \(ab\)?
by
Upanshu Gupta
Evaluate the integral \(\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^2+2x+4} \, dx.\)
Round your answer to three decimal places.
by
jatin yadav
Evaluate:
\[\large \displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cot(x)\sec^2(x)}{\cot(x) + 1}\, dx\]
by
Andrew Ellinor