1.2 Problems
25
1.2.6 Series
1.35 Find the interval of convergence for the series:
x −
x
2
2 2 +
x
3
3 2 −
x
4
4 2 + · · ·
1.36 Expand log x in powers of (x − 1) by Taylor’s series.
1.37 Expand cos x into an infinite power series and determine for what values of x
it converges.
1.38 Expand sin(a + x) in powers of x by Taylor’s series.
1.39 Sum the series s = 1 + 2x + 3x
2
+ 4x
3
+ · · · , |x| < 1
1.2.7 Integration
1.40 (a) Evaluate the integral:
sin
3 x cos
6 x dx
(b) Evaluate the integral:
sin
4 x cos
2 x dx
1.41 Evaluate the integral:
1
2x 2 − 3x − 2
dx
1.42 (a) Sketch the curve in polar coordinates r
2
= a
2 sin 2θ
(b) Find the area within the curve between θ = 0 and θ = π/2.
1.43 Evaluate:
(x
3
+ x
2
+ 2)
(x 2 + 2) 2 dx
1.44 Evaluate the definite integral:
+∞
0
4a
3
x 2 + 4a 2 dx
1.45 (a) Evaluate:
tan
6 x sec
4 x dx
(b) Evaluate:
tan
5 x sec
3 x dx
25
1.2.6 Series
1.35 Find the interval of convergence for the series:
x −
x
2
2 2 +
x
3
3 2 −
x
4
4 2 + · · ·
1.36 Expand log x in powers of (x − 1) by Taylor’s series.
1.37 Expand cos x into an infinite power series and determine for what values of x
it converges.
1.38 Expand sin(a + x) in powers of x by Taylor’s series.
1.39 Sum the series s = 1 + 2x + 3x
2
+ 4x
3
+ · · · , |x| < 1
1.2.7 Integration
1.40 (a) Evaluate the integral:
sin
3 x cos
6 x dx
(b) Evaluate the integral:
sin
4 x cos
2 x dx
1.41 Evaluate the integral:
1
2x 2 − 3x − 2
dx
1.42 (a) Sketch the curve in polar coordinates r
2
= a
2 sin 2θ
(b) Find the area within the curve between θ = 0 and θ = π/2.
1.43 Evaluate:
(x
3
+ x
2
+ 2)
(x 2 + 2) 2 dx
1.44 Evaluate the definite integral:
+∞
0
4a
3
x 2 + 4a 2 dx
1.45 (a) Evaluate:
tan
6 x sec
4 x dx
(b) Evaluate:
tan
5 x sec
3 x dx