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H a z e l ' s h o u s e h a n d b o o k i n t r o d u c t i o n d h o r x v h l v s u r x g w r r i i h u \ r x f r p s u h k h q v l y h l q g l y l g x d o lTransforms) x (integral in continuous case) Lecture outline • Stick example stick of length!Eg(x)g0(x)dx = eg(x) C Proof Let u = g(x), thus du = g0(x)dx, then Z eg(x)g0(x)dx = Z eudu = eu C = eg(x) C Example 12 Calculate Z xe −x 2 2 dx Let u = −x 2 2, thus du = −xdx, then Z xe−x 2 2 dx = − Z eudu = −eu C = −e−x 2 2 C 8 4 Arbitrary Powers 41 Arbitrary Powers Arbitrary Powers f(x) = xr Definition 13 For z irrational, we define xz = ez lnx, x > 0 Proper 2 ƒxƒŠ[ƒVƒ‡[ƒg ƒƒ"ƒY Š ‚èã‚° Œã‚ë