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6. Integration 6.2 Substitution Why Example 1. Getting acquainted: Find the following indefinite integral Z x5 (x6 + e7 )8 dx How Based on the reverse chain rule. Recall of chain rule: Let u = g(x) = x6 + e7 . Then, the differential of u du = So, dx = Thus, Z x5 (x6 + e7 )8 dx 1 Check (by differentiation as we did in 6.1): What The Substitution Rule: Let u = g(x), we have Z Z 0 f (g(x))g (x) dx = f (u) du, assuming that u = g(x) is a continuously differentiable function whose range is an interval on which f is continuous. Trick Choosing the right u in the integrand. It depends on the integral. Practice for gaining experience. SOME PRACTICE. Finding the following indefinite integrals. Example 2. Z x(−x2 + 9)2014 dx Example 3. Z √ x 2 + 5x2 dx 2 Example 4. Z x − ln2 dx (x2 − 2xln2)2 Example 5. x − ln2 dx − 2xln2 Z x2 Example 6. Z e2x dx ex + e 3 Example 7. Z (1 − x)2 ex 3 −3x2 +3x−1 Example 8. Z 2x √ dx 3 x−π 4 dx Example 9. Z provided Z √ 1 x2 + 1 1 √ dx, 1 + 9x √ dx = ln x + x2 + 1 + C. Example 10. Z r x dx 2−x 5 Example 11. Z √ 1 − x2 dx More examples 6