Integration by substitution "Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an integral, but only when it can be set up in a special way.[br]The first and most vital step is to be able to write our integral in this form:[br][br][img width=173,height=55]https://lh3.googleusercontent.com/4Mj8y6LGaxQNGUdXKPYj-1DORX09ATXR6UiRbM02-uQxLh4B_S5npGEu3Wv5FjkZtFeUs_gQzxN6AgHQGCkHYnPM4PGSIrwg6aYodOxpgwUB44L29d-uROEwbLHQT8o5xxaswFOa[/img][br][br]Note that we have g(x) and its derivative g'(x).[br][br]Like in this example:[br][br][br][img width=166,height=51]https://lh4.googleusercontent.com/Z4Gf0JF8sC4dRv1o5qjyDl6P84pCxRpolw1AmlDBQI1kTeCDNAiKniVY2HARdF6wsz_8uuS-iweoBfMGivb2_sTrOzyZUNV8B9C7uS28NbzfutRrN2faUaP-FjVp9DP3AbzDLxjI[/img][br][br]Here f=cos, and we have g=x2 and it's derivative 2xThis integral is good to go![br]When our integral is set up like that, we can do this substitution:[br][br][img width=185,height=75]https://lh3.googleusercontent.com/EH8I3FT-Zq-of7wXqUFy4nbEY0ubsHBJCL8i5eheRalR65difQU4gNsmtG1txLhjoc1jiagChALy4GNmu-A63ygQMCnLvUwMGImKqqhg95l7aQre9EAujT59IEPcklpC2SxC4Z3K[/img][br][br]Then we can integrate f(u), and finish by putting g(x) back as u.We know (from above) that it is in the right form to do the substitution:[br][br][br][img width=181,height=75]https://lh4.googleusercontent.com/Itzzp53LfWMlrz60nosu1Rzvsu3htMgDLYyEY2gw5hFw80ilnC2mW_cmYTKFDoFuPjr0TrDGyckVrWeYzKMTovHETNvAHqnltNAqSv_kb5M8N5puMs8XJVds-t606kXtRw2LEXxh[/img][br][br][br]Now integrate:∫cos(u) du = sin(u) + CAnd finally put u=x2 back again:sin(x2) + C  (Math is Fun, n.d)[br][br][br]