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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.

\[\large\int_{0}^{2\pi}\cos (lx)\cos (mx)\cos (nx) \, dx \]

Let \(l,m,n\) be integers such that their product is non-zero. Evaluate the integral above.

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

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 \(ab\)?

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Evaluate the integral : \(\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^2+2x+4} \, dx\)

- \(\ln x \) is the natural logarithm.
- Round your answer to three decimal places.

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Evaluate:

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

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