How would you prove the 'integration by parts' rule?
This involves thinking about a well-known formula (the product rule) in a slightly different way. Looking at the product rule, for two functions u and v, (uv)' = uv' + vu'. We can rewrite this as uv' = (uv)' - vu'. Integrating both sides, we obtain integral of uv' = uv - integral of vu'.
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All answers ▸Differentiate: f(x)=(ax^2 + bx + c) ln(x + (1+x^2)^(1/2)) + (dx + e) (1 + x^2)^(1/2). Hence integrate i) ln(x + (1 + x^2)^(1/2)), ii) (1 + x^2)^(1/2), iii) x ln(x + (1 + x^2)^(1/2)).
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