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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.

Level 3

         

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

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

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

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

Evaluate:

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

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