solve problems calculus 1

Basic Antiderivative Rules Math 2414

Summer I, 2009

Definition A function F is an antiderivative of the function f on an interval I if F ′(x) = f (x) for every x in I. The notation∫

f (x) dx = F (x) + C,

where F ′(x) = f (x) and C is an arbitrary constant, denotes the family of all antideriva- tives of f (x) on an interval I.

Basic Antiderivative Rules

1. ∫

k dx = kx + C

2. ∫

kf (x) dx = k ∫

f (x) dx

3. ∫

[f (x) ± g(x)] dx = ∫

f (x) dx ± ∫

g(x) dx

4. ∫

xn dx = xn+1

n + 1 + C, n 6== 1

5. ∫

sin(x) dx = −cos(x) + C

6. ∫

cos(x) dx = sin(x) + C

7. ∫

sec2(x) dx = tan(x) + C

8. ∫

csc2(x) dx = −cot(x) + C

9. ∫

sec(x) tan(x) dx = sec(x) + C

10. ∫

csc(x) cot(x) dx = −csc(x) + C

11. ∫

ex dx = ex + C

12. ∫

1

x dx = ln |x| + C

Basic Antiderivative Rules Math 2414

Summer I, 2009

Evaluate each indefinite integral.

1. ∫

(x + 3) dx

2. ∫ (

x3 + 5 )

dx

3. ∫ (

x3/2 + 2x + 1 )

dx

4. ∫

1

x3 dx

5. ∫

x2 + x + 1 √

x dx

6. ∫

(x + 1)(3x − 2) dx

7. ∫

y2 √

y dy

8. ∫

[2 sin(x) + 3 cos(x)] dx

9. ∫

[1 − csc(t) cot(t)] dt

10. ∫ [

tan2(y) + 1 ]

dy

11. ∫ (

x − 5

x

) dx

12. ∫

x3 − 1 x − 1

dx

13. ∫

x3 + 3x2 − 9x − 2 x − 2

dx

14. ∫

sec(x) sin(x)

cos(x) dx

15. ∫

[1 + cot2(x)] cot(x)

csc(x) dx