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- 1. Auntie Derivative's Party!
- 2. Here we can apply substitution where u = ln( z ). But sometimes, as in this case, it might be necessary to solve for z in terms of u , depending on the complexity of the integrand.
- 3. The variables u and du can now be substituted into the integrand, but now the z must be converted into u so antidifferentiation can occur.
- 4. Since z = e , substituting this into the integrand yields an expression only in terms of u . u
- 5. Now integration by parts can be applied to the variable u . Don't forget to use LIATE.
- 6. Here's LIATE again in case anyone has forgotten. Remember, it's purpose is to help determine f. L - Logarithmic (ex. log( x )) I - Inverse Trig. (ex. arccos( x )) A - Algebraic (ex. x 2 ) T - Trigonometry (ex. cos( x )) E - Exponential (ex. e x )
- 7. After applying integration by parts, it's obvious that it's still not simple enough to solve, so we must use parts again.
- 8. Now the full expression can be simplified and e u can easily be antidifferentiatied.
- 9. Remember, u = ln( z ), so once the full expression is simplified you must resubstitute ln( z ) in. Now the final answer is known.
- 10. This approach involves u-substitution as the first step, where u = x 3 .
- 11. We can take out the e as a coefficient for the antiderivative, but notice how u has been substituted into the integrand twice where x 3 is present. Two substitutions.
- 12. Now that the the integrand is only in terms of u , we can now antidifferentiate using parts.
- 13. Once again using LIATE, f and g' can be determined for the parts.
- 14. All that is left is to finish antidifferentiating f'g . Also, don't forget that e /3 is to be distributed to the whole antiderivative, thus the brackets.
- 15. Now that there are no indefinite integrals left in the expression, all that is left is some algebraic massage. In this step, I have already factored out 2/3.
- 16. You can factor out a ( u + 1) 3/2 to further simplify the expression. Don't forget that resubstituting into u is now required.
- 17. Here is the final answer, though further algebraic massage is possible, the answer will still remain the same function.
- 18. LIATE indicates that algebraic functions have priority over trigonometric functions for f , as used here.
- 19. The antiderivative of tan( x ) must now be determined, which isn't a fundamental antiderivative.
- 20. But don't forget that we always have our trigonometric functions.
- 21. Define the function for substitution.
- 22. Applying the variable u to the integrand works nicely.
- 23. Don't forget to distribute that negative sign and also to subsitute u = cos( x ) once you antidifferentiate.
- 24. Since the antiderivative of tan( x ) has now been determined, the full antiderivative of x sec 2 ( x ) can easily be determine as above.
- 25. Well, that's it for thursday's scribe. I hope that anyone who didn't understand what was going on during thursday's class when Mr. K was going through these tougher questions found my dissection helpful. Goodbye for now! The next scribe is: MrSiwWy! =D

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