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Integral x*sqrt(x-1). Apply integration by parts, square root example. Choose u=x!
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For the integral x*sqrt(x-1), a simple u-substitution of u=x-1 would be easier, but we illustrate how to apply integration by parts. For the integral of a product of two functions, we apply integration by parts, and in this case it's integration by parts with fractional power of 1/2. We have to choose u and dv, and we choose dv=sqrt(x-1)dx and choose u=x. In the integration by parts formula, we have to pick u and dv, and the trick is to choose u so it's derivative is simpler than the u we started with, so this is how to choose u for integration by parts. As long as the antiderivative of dv is guessable and the derivative of u gets simpler, it's a good setup.
Next, we apply the integration by parts formula and we obtain a new expression with an integral whose antiderivative is guessable. We clean things up for the result of integration by parts and we're done!
To compare the answers from both approaches (u-substitution vs. integration by parts), we would have to simplify the final answers quite a bit, but they do come out the same.
Next, we apply the integration by parts formula and we obtain a new expression with an integral whose antiderivative is guessable. We clean things up for the result of integration by parts and we're done!
To compare the answers from both approaches (u-substitution vs. integration by parts), we would have to simplify the final answers quite a bit, but they do come out the same.
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