1.2 Problems 25
1.2.6 Series............................................
1.35 Find the interval of convergence for the series:
x−x^2
22+
x^3
32−
x^4
42+···
1.36 Expand logxin powers of (x−1) by Taylor’s series.
1.37 Expand cosxinto an infinite power series and determine for what values ofx
it converges.
1.38 Expand sin(a+x)inpowersofxby Taylor’s series.
1.39 Sum the seriess= 1 + 2 x+ 3 x^2 + 4 x^3 +···,|x|< 1
1.2.7 Integration........................................
1.40 (a) Evaluate the integral:
∫
sin^3 xcos^6 xdx
(b) Evaluate the integral:
∫
sin^4 xcos^2 xdx1.41 Evaluate the integral:
∫
1
2 x^2 − 3 x− 2
dx1.42 (a) Sketch the curve in polar coordinatesr^2 =a^2 sin 2θ
(b) Find the area within the curve betweenθ=0 andθ=π/2.
1.43 Evaluate:
∫
(x^3 +x^2 +2)
(x^2 +2)^2
dx1.44 Evaluate the definite integral:
∫+∞
04 a^3
x^2 + 4 a^2dx1.45 (a) Evaluate:
∫
tan^6 xsec^4 xdx
(b) Evaluate:
∫
tan^5 xsec^3 xdx