Calculus2

The Substitution Rule

� 1, 2 Evaluate the integral by making the given substitution.

1.

∫ x2

√ x3 + 1 dx, u = x3 + 1

2.

∫ esin θ cos θdθ, u = sin θ

� 3–10 Evaluate the indefinite integral.

3.

∫ 2x(x2 + 3)4 dx 4.

∫ √ 4 + 3xdx

5.

∫ (ln x)2

x dx 6.

∫ yey

2

dy

7.

∫ x

x2 + 1 dx 8.

∫ x

√ 1 − 4x2

dx

9.

∫ sin θ

1 + cos2 θ dθ 10.

∫ t

1 + t4 dt

� 11–14 Evaluate the definite integral.

11.

∫ 2 1

1

(3 − 5x)2 dx 12.

∫ π 0

θ cos(θ2) dθ

13.

∫ a 0

x √ a2 −x2 dx 14.

∫ 2 1

x √ x− 1 dx

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15. The antiderivatives of sin kx and cos kx are worth remembering.

a) Use substitution to find a formula for ∫

sin kxdx.

b) Differentiate the result to confirm the procedure.

16. A colony of bacteria cells is placed into a petri dish and grows at a rate of 1.7e.085t cells per hour. Find the net change in the colony’s size over the first 24 hours.

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Solutions to Selected Problems

1. 2

9 (x3 + 1)

3/2 + C

2. esin θ + C

3. 1

5 (x2 + 3)5 + C

4. 2

9 (4 + 3x)

3/2 + C

5. 1

3 (ln x)3 + C

6. 1

2 ey

2

+ C

7. ln(x2 + 1)

2 + C

8. − √

1 − 4x2 4

+ C

9. −arctan(cos θ) + C

10. arctan t2

2 + C

11. 1

14

12. sin π2

2

13. a3

3

14. 16

15

15. a) − 1

k cos(kx) + C

b) Confirm

16. ≈ 133.8 cells

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