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Calculus/Calculus I (Substitution Rule and Definite Integrals)

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Expert: - 6/26/2008


Question
Please help! I'm not sure how to do this one and I must have it done by tomorrow (6-27-08)! I was able to do all of the indefinite integrals but this one is a definite and I am confused. Please give me step by step of how to do this.

Question: Evaluate the definite integral    ∫[0,4] ((x)/(sqrt(1+2x)))dx

(Note: The interval [0,4] follows the integral sign directly with the 4 being above the 0)

***I have attempted this problem and gotten (1/2)∫[1,9]  ((x)/(sqrt(u)))du by determining that u= 1+2x, du= 2dx, and du/2= sx as well as u=1 when x=0 and u=9 when x=4  (which are my new intervals). Please let me know if I was on the right track!!! I really appreciate it and hope you can help.  

Answer
Have you ever done ∫u dv = uv - ∫v du?

Let u=x, then du=dx.
Let dv = 1/(sqrt(1+2x)) dx, so v = sqrt((1+2x) since the half and the two cancel in the derivative.

The variables u and v are now both known, so all that has to be done is to integrate v.  The variable v is just (1+2x)^0.5.  Don't forget to worry about d(2x)/dx=2 when doing this.

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