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Drill: Find dy / dx. y = x 3 sin 2x y = e 2x ln (3x + 1) y = tan -1 2x Product rule: x 3 (2cos 2x) + 3x 2 sin (2x) 2x 3 cos 2x + 3x 2 sin (2x). Product Rule e 2x (3/(3x +1)) + 2e 2x ln (3x + 1) 3e 2x /(3x +1) + 2e 2x ln (3x + 1). Antidifferentiation by Parts. Lesson 6.3.

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drill find dy dx
Drill: Find dy/dx
  • y = x3 sin 2x
  • y = e2xln (3x + 1)
  • y = tan-1 2x
  • Product rule:
  • x3 (2cos 2x) + 3x2 sin (2x)
  • 2x3cos 2x + 3x2 sin (2x)
  • Product Rule
  • e2x (3/(3x +1)) + 2e2xln (3x + 1)
  • 3e2x/(3x +1) + 2e2xln (3x + 1)
objectives
Objectives
  • Students will be able to:
    • use integration by parts to evaluate indefinite and definite integrals.
    • use rapid repeated integration or tabular method to evaluate indefinite integrals.
integration by parts formula
Integration by Parts Formula

A way to integrate a product is to write it in the form

If u and v are differentiable function of x, then

example 3 solving an initial value problem
Example 3 Solving an Initial Value Problem
  • Solve the differential equation dy/dx = xlnx subject to the initial condition y = -1 when x = 1

It is typically better to let u = lnx

drill
Drill

Solve the differential equation: dy/dx= x2e4x (This means you will need to find the anti-derivative of dy/dx = x2e4x)

rapid repeated integration by parts aka the tabular method
Rapid Repeated Integration by PartsAKA: The Tabular Method
  • Choose parts for u and dv.
  • Differentiate the u’s until you have 0.
  • Integrate the dv’s the same number of times.
  • Multiply down diagonals.
  • Alternate signs along the diagonals.
homework
Homework
  • Page 346/7: Day #1: 1-15 odd
  • Page 347: 17-24
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