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Integrals of Exponential and Logarithmic Functions

Integrals of Exponential and Logarithmic Functions. Integration Guidelines. Learn your rules (Power rule, trig rules, log rules, etc.). Find an integration formula that resembles the integral you are trying to solve ( u -substitution should accomplish this goal ).

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Integrals of Exponential and Logarithmic Functions

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  1. Integrals of Exponential and Logarithmic Functions

  2. Integration Guidelines • Learn your rules (Power rule, trig rules, log rules, etc.). • Find an integration formula that resembles the integral you are trying to solve (u-substitution should accomplish this goal). • If u-substitution does not work, you may need to alter the integrand (long division, factor, multiply by the conjugate, separate the fraction, or other algebraic techniques). • When all else fails, use your TI-89.

  3. Power Rule for Integrals for n ≠ -1 The power rule is not valid for n = -1 because: (meaningless)

  4. Derivative and Antiderivatives that Deal with the Natural Log However, we know the following to be true: This shows that lnx is an antiderivative of 1/x. Or: But, thus formula is only valid for x>0 (where lnx is defined). How can we have an antiderivative on its full domain?

  5. Indefinite Integral of y = 1/x If x ≠ 0, then This is valid because we proved the following result:

  6. Derivative and Antiderivatives that Deal with the Exponentials We know the following to be true: Solve for ax: (Constant Rule in reverse) This shows the antiderivative of ax : As long as a>0 (where lna is defined), this antiderivative satisfies all values of x.

  7. Indefinite Integral of Exponentials If a > 0, then If a = e, then

  8. Example 1 Define u and du: Substitute to replace EVERY x and dx: Solve for dx Integrate. Substitute back to Leave your answer in terms of x. Find .

  9. Example 2 Define u and du: Substitute to replace EVERY x and dx: Solve for dx Integrate. Substitute back to Leave your answer in terms of x. Find for x>1.

  10. Example 3 Do not forget about all of your old techniques Find .

  11. Example 4 Find x 2 Rm Perform Polynomial Division. x x2 2x 4 - 3 -3x -6 x2 – x – 2 Thus:

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