4 .9 Antiderivatives

1. 4.9 Antiderivatives

2. 1. Definition: F(x) is an antiderivative of f(x) if F′(x)=f(x). 2. Example: f(x)=x3 +2x F1 (x)= F2 (x)= F (x)=

3. 3. Theorem: If F(x) is an antiderivative of f(x), then so is F(x)+C.

4. Antiderivative xn+1 ———+C n+1 sin(x) -cos(x)+C cos(x) sin(x)+C sec2 (x) tan(x)+C sec (x)tan(x) sec(x)+C

5. Antiderivative Cx+D C Cf(x) CF(x) f(x)±g(x) F(x)±G(x) Ln |x|+C x-1 ex ex +C

6. Example 1: f(x)=sin(x)+x2. Find the most general antiderivativeof f. x 3 Solution: F(x)= -cos(x)+—+C 3 Check:

7. x 4 +x5 +2 f (x)= ————. Ex 2. Find the most x2 general antiderivative of f. Solution:

8. Check:

9. 1 Example 3: If f′(x)=1+— x2 and f(1)=2. Find f(x). Solution: f(x)= f(1)= f(x)=

10. Example 4: f′(x)=4 - sec(x) tan(x) Find f′(x). f′(x)= sec(x)+ Solution: + = +sec(x)+ C

11. Example 5:f″(x)= x2 +3 sin(x), f(0)=2, f′(0)=3,. Find f(x). Solution: f′(x)= 3 cos(x) + C f′(x)= 3 3(1)+ f′(x)= 3 cos(x) + 6 f(x)= 3 sin(x)+

12. f(x)=

13. , f(1)=1, f(2)=0. Find f(x). Example 6: f″(x)= Solution: f′(x)= f(x)= f(1)= f(2)= 0 ....(2)

14. Subtract (2) from (1), we get Substitute into (2) for C1, we get

15. Subtract (2) from (1), we get Substitute into (2) for C2, we get f(x)=