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Int by Parts Notes

# Int by Parts Notes - Z 3 2 ln xdx Example 0.3 Using...

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Integration by Parts (Section 7.1) Looking at the product rule for derivatives: d dx ( uv ) = u dv dx + du dx v Integrate to get: We rearrange to get the integration by parts formula: Z udv = uv - Z vdu For definite integrals: Z b a udv = uv | b a - Z b a vdu Given an integral, how do we choose u ?

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Example 0.1. Evaluate Z x cos xdx . 1. Choose u . The remaining part of the integral is dv . 2. Find du and v . 3. Use formula and evaluate. **It’s nice when you can find a u that will ”disappear” when you take du . ** dv needs to be something you can find the antiderivative of. Example 0.2. Evaluate

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Unformatted text preview: Z 3 2 ln xdx . Example 0.3. Using integration by parts to evaluate integral of inverse trig functions: Evaluate Z tan-1 xdx . Example 0.4. Making a substitution ﬁrst: Evaluate Z π 2 / 16 sin √ xdx . Example 0.5. Integrating by parts more than once: Evaluate Z x 2 e 2 x dx . Example 0.6. What if neither u nor v disappear? Evaluate Z e x cos xdx . Practice in groups 1. Z 3 xe 2 x dx 2. Z e 1 x 2 ln xdx 3. Z e x ln( e x ) dx 4. Z sin-1 xdx 5. Z e 2 y cos(3 y ) dy...
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