Higher Odd Powers of Secant
Just as the integration by parts above reduced the integral of secant cubed to the integral of secant to the first power, so a similar process reduces the integral of higher odd powers of secant to lower ones. This is the secant reduction formula, which follows the syntax:
Alternatively:
Even powers of tangents can be accommodated by using binomial expansion to form an odd polynomial of secant and using these formulae on the largest term and combining like terms.
Read more about this topic: Integral Of Secant Cubed
Famous quotes containing the words higher, odd and/or powers:
“Painting seems to be to the eye what dancing is to the limbs. When that has educated the frame to self-possession, to nimbleness, to grace, the steps of the dancing-master are better forgotten; so painting teaches me the splendor of color and the expression of form, and as I see many pictures and higher genius in the art, I see the boundless opulence of the pencil, the indifferency in which the artist stands free to choose out of the possible forms.”
—Ralph Waldo Emerson (1803–1882)
“Isn’t it odd that networks accept billions of dollars from advertisers to teach people to use products and then proclaim that children aren’t learning about violence from their steady diet of it on television!”
—Toni Liebman (20th century)
“All the powers of imagination combine in hypochondria.”
—Mason Cooley (b. 1927)