Calculus2

Integration by Parts

� 1, 2 Do the following:

a) Evaluate the integral using the indicated choices of u and dv.

b) Confirm your answer by differentiation.

1.

∫ x ln x dx, u = ln x, dv = x dx

2.

∫ θ cos θ dθ, u = θ, dv = cos θ dθ

� 3–8 Use Parts to evaluate the integral.

3.

∫ xe

−x dx 4.

∫ t sin 2t dt

5.

∫ p 5 ln p dp 6.

∫ 2

1

ln y

y2 dy

7.

∫ 1/2

0

sin−1 x dx 8.

∫ e 1

(ln x)2 dx

� 9, 10 First make a t-substitution, and then use Parts to evaluate the integral.

9.

∫ θ 5 cos(θ3) dθ

10.

∫ 4

1

e √

x dx

� 11, 12 Reduction formulas are used to “reduce” an integral involving a power to an integral of lower power. Consider the reduction formula

∫ (ln x)n dx = x(ln x)n − n

∫ (ln x)n−1 dx (1)

11. Use Parts to prove Equation 1.

12. Use Equation 1 to find ∫ (ln x)3 dx.

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Solutions to Selected Problems

1. x2 ln x

2 −

x2

4 + C

2. θ sin θ + cos θ + C

3. −(x + 1)e−x + C

4. sin(2t) − 2t cos(2t)

4 + C

5. p6 ln p

6 −

p6

36 + C

6. 1 − ln 2

2

7. π + 6

√

3

12 − 1

8. e − 2

9. θ3 sin θ3 + cos θ3

3 + C

10. 2e2

11. Let u = (ln x)n and dv = 1dx

12. x(ln x)3 − 3x(ln x)2 + 6x ln x − 6x + C

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