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| Integration by Parts |
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| Let u and v be two differentiable function of a single independent variable x. |
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| Integrating w.r.t x on both sides |
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Let u = f(x),  |
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Then  |
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| \ (1) can be written as |
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| Note: |
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| The above can be remembered as |
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 (First function) (Integral of Second function) |
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| Note: |
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| (1) While integration by parts, the proper choice of first function and second function is significant |
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Try integrating  |
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| Taking x as the first function and ex as the second function is simpler than taking x as the second function. |
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| (2) Integrating by parts may not be applicable to product of functions in all cases. In some cases the product of two functions may not be integrable. |
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| (3) While finding the integral of the second function we do not add constant of integration. |
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| We need not add a constant of integration to the second function as it gets cancelled in the final results. |
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| (4) Some times even if the integral is not a product of two functions, the method of integration by parts can be used. |
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| Example: Let us integrate sin-1x |
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where   |
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| Example: |
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| Integrate the following function (x2 + 1) logx |
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| Suggested answer: |
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| Integrating by parts |
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| Method: |
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 Put ex f(x) = t |
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| Differentiating w.r.t. x, |
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| {exf(x) +exf'(x)} dx = dt |
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| ex{f(x) + f'(x)} dx = dt |
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| = t + c |
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| = exf(x)+c |
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Evaluate the integral  . |
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| Suggested answer: |
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| Differentiating, |
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