Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 38
∫
(q) e x√ ∫
1+e x dx (r) xe x2 +1 dx
∫
∫
dx
4
(s)
(t)
xlnx
x(lnx+1) dx 2
∫ √cotx
∫
(u) csc 2 xdx (v) sinx(cosx−1) 3 dx
∫
∫ e
(w) sec 3 x −e −x
xtanxdx (x) dx
e x +e−x ∫ ∫ 1+x
2x+3
(y)
1+x dx (z) 2 x+7 dx
Answer to Exercise 4.2
1. (a) 1 3 sin3x+C (b) 2 3 (x3 +2) 3/2 +C (c) −1/(1+2x) 2 +C (d) 1 2 (√ x+2) 4 +C
2. (a) 1 5 (x2 +3) 5 +C (b) 1 4 (x2 +x) 4 +C (c) 2 3 (x−1)3/2 +C (d) 2 3√
x3 −2+C
(e) − 1 3 ln|5−3x|+C (f) ln|x2 +x−1|+C (g) 2 √ 1+x+2x 2 +C
(h) −2( √ x+1) −1 +C (i) 1 2 sin2x+C (j) 2 3 (sinx+1)3/2 +C (k) − 1 2 cos(x2 )+C
(l) −2 √ cosx+C (m) − 1 5 cos5 x+C (n) − 4 7 (cosx+3)7/4 +C
(o) 2 3 (1+secx)3/2 +C (p) e sinx +C (q) 2 3 (1+ex ) 3/2 +C (r) 1 2 ex2 +1 +C
(s) ln|lnx|+C (t) −4(lnx+1) −1 +C (u) − 2 3 (cotx)3/2 +C (v) − 1 4 (cosx−1)4 +C
(w) 1 3 sec3 x+C (x) ln(e x +e −x )+C (y) tan −1 x+ 1 2 ln(1+x2 )+C
(z) 2(x+7)−11ln|x+7|+C
Exercise 4.3
1. Express the limit as a definite integral on the given interval.
(a) lim
n→∞
n∑
i=1
x i sinx i △x, [0,π]
n∑ e x i
(b) lim △x, [1,5]
n→∞ 1+x
i=1 i
(c) lim
n∑
n→∞
i=1
n∑
n→∞
(d) lim
i=1
[2(x ∗ i) 2 −5x ∗ i]△x, [0,1]
√
x
∗
i △x, [1,4]