Integration

1. Integration Substitution Method

2. Please integrate … You Can’t … at least not now, right?! There are several integration techniques we can employ … the simplest of these is the substitution method. The key to substitution: the integrand must contain a function and its derivative in order to work. On the next slide we’ll see how it works.

3. Notice how the integrand contains both a function and its derivative … Here’s how substitution works …define the initial terms If we let u = sin x take derivatives on both sides then du = cos x dx now let’s substitute OK … now we’re done … evaluate … Rewrite the upper and lower bounds in terms of “u” u = sin(π/2) = 1 u = sin(0) = 0

4. Here’s another example … find a function and its derivative The terms of substitution let then To see it a little more clearly, lets move terms around in the integrand. Let’s substitute …. Notice how the limits of integration changed.

5. Integration using substitutionExample 2 Integrate Now substitute back

6. Integration using substitutionExample 3 • Use the substitution u = 5 – x2 to find • Differentiating u =5-x2 gives • Changing the variable gives • We now have • which we can integrate

7. Integration using substitutionExample 3 Now substitute back for u

8. Oh dear we have a bit left over Integration using substitutionExample 4 • Use the substitution u = 2x +1 to find • Differentiating u = 2x +1gives • Changing the variable gives • We now have

9. Integration using substitutionExample 4 - continued When we have a bit left over ….. Since, u = 2x +1 We can rearrange to: x = (u – 1)/2 We can rewrite the integral as …… Now substitute back for u