4.5 Integration by Substitution Recall the Chain Rule. We used this when we had a function being _________________________________________________. Take the derivative of f (x) = (x 2 + 1)3 Integration by substitution is a way that we can ______________________________________ for integration. Look at 2
2
∫ x(x + 1) dx Just as when we used Chain Rule in derivative there is a function being ACTED ON BY ANOTHER FUNCTION. To integrate this we are going to ALWAYS make the function being acted on by another function u, and the derivative of that function du. We must now check to see that the integral has all the “parts” necessary to find the antiderivative. Try ∫ x 2 x 3 − 2 dx BE SURE TO FIRST ID THE FUNCTION ACTED ON BY ANOTHER FUNCTION Change of Variables ∫ x 2x − 1 dx Just as before ID u and du. What is different this time? ∫ sin
2
x cos x dx 3
3x cos 3x dx ∫ sin