Math 162

Class 8

\(\displaystyle \int_1^2 x^2\exp(2x)\;dx\)

\(\displaystyle \int \ln(x^2)\;dx\)

\(\displaystyle \int_1^{\exp(\pi)} \cos(2\ln x)\;dx\)

Important Trigonometric Formulas

• Pythagorean Identities

  • \(\cos^2 t + \sin^2 t = 1\)
  • \(1 + \tan^2 t = \sec^2 t\)
  • \(\cot^2 t + 1 = \csc^2 t\)

• Power reduction identities:

  • \(\displaystyle \cos^2 t = \frac{1 + \cos(2t)}{2}\)
  • \(\displaystyle \sin^2 t = \frac{1 - \cos(2t)}{2}\)

\(\displaystyle\int_0^{\pi/2} \sin(x)\cos(x)\;dx\)

\(\displaystyle\int_0^{\pi/2} \sin^2(x)\cos(x)\;dx\)

\(\displaystyle\int_0^{\pi/2} \sin^2(x)\cos^3(x)\;dx\)

\(\displaystyle\int_0^{\pi/2} \cos^2(x)\;dx\)

\(\displaystyle\int_0^{\pi/2} \cos^4(x)\sin^5(x)\;dx\)

\(\displaystyle\int_0^{\pi/2} \cos^2(x)\sin^4(x)\;dx\)