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.
1 of 2
Solution
s 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
2 of 2
...
Integration by Parts 1, 2 Do the followinga) Evaluat.docx
1. 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.
2. ∫
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
3. 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.
1 of 2
4. Solution
s 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
5. + C
5.
p6 ln p
6
−
p6
36
+ C
6.
1 − ln 2
2
7.
π + 6
√
3
6. 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
2 of 2
