Exercise 3.3
Exercise 3.3 Exercise 3.3
MA111: Prepared by Dr.Archara Pacheenburawana 37 Answer to Exercise 4.1 1. (a) 5x+C (b) 1 3 x3 +πx+C (c) 4 9 x9/4 +C (d) 3 3√ x+C (e) 1 3 x3 − 1 2 x2 +C (f) 2 3 x6 − 1 4 x4 +C (g) 27 8 x8 + 1 2 x6 − 45 4 x4 + √ 2 2 x2 +C (h) − 3 x + 1 x 2 +C (i) x 4 + 3 2 x2 +C 2. (a) 3 5 x5 +C (b) 1 3 x3 + x2 2 +C (c) 3 5 x5 − 3 2 x2 +C (d) 1 3 (x+1)3 +C (e) 2x 3/2 +C (f) 3x+ 1 3 x−3 +C (g) 3 2 x2/3 −9x 1/3 +C (h) 2 9 x9/2 + 4 5 x5/2 +2x 1/2 +C (i) −cosx−sinx+C (j) 2secx+C (k) 5tanx+C (l) 3e x −2x+C (m) 3sinx−ln|x|+C (n) 5 2 x2 +3e −x +C (o) x−3e −x +C (p) 2 5 x5/2 − 16 5 x5/4 Exercise 4.2 1. Evaluate the integral by making the given substitution. ∫ (a) cos3xdx, u = 3x ∫ (b) x 2 (x 3 +2)dx, u = x 3 +2 ∫ 4 (c) dx, u = 1+2x (1+2x) 3 ∫ √ ( x+2) 3 (d) √ dx, u = √ x+2 x 2. Evaluate the indefinite integral ∫ (a) 2x(x 2 +3) 4 dx ∫ √x−1dx (c) ∫ (e) ∫ (g) dx 5−3x 1+4x √ dx 1+x+2x 2 ∫ (i) cos2xdx ∫ (k) xsin(x 2 )dx ∫ (m) cos 4 xsinxdx ∫ (o) secxtanx √ 1+secxdx ∫ (b) (2x+1)(x 2 +x) 3 dx ∫ x 2 (d) √ x3 −2 dx ∫ (f) ∫ (h) 2x+1 x 2 +x−1 dx 1 √ √ dx x( x+1) 2 ∫ (j) cosx √ sinx+1dx ∫ sinx (l) √ dx cosx ∫ (n) sinx(cosx+3) 3/4 dx ∫ (p) cosxe sinx dx
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]
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MA111: Prepared by Dr.Archara Pacheenburawana 37<br />
Answer to <strong>Exercise</strong> 4.1<br />
1. (a) 5x+C (b) 1 3 x3 +πx+C (c) 4 9 x9/4 +C (d) 3 3√ x+C (e) 1 3 x3 − 1 2 x2 +C<br />
(f) 2 3 x6 − 1 4 x4 +C (g) 27<br />
8 x8 + 1 2 x6 − 45 4 x4 + √ 2<br />
2 x2 +C (h) − 3 x + 1 x 2 +C<br />
(i) x 4 + 3 2 x2 +C<br />
2. (a) 3 5 x5 +C (b) 1 3 x3 + x2<br />
2 +C (c) 3 5 x5 − 3 2 x2 +C (d) 1 3 (x+1)3 +C<br />
(e) 2x 3/2 +C (f) 3x+ 1 3 x−3 +C (g) 3 2 x2/3 −9x 1/3 +C (h) 2 9 x9/2 + 4 5 x5/2 +2x 1/2 +C<br />
(i) −cosx−sinx+C (j) 2secx+C (k) 5tanx+C (l) 3e x −2x+C<br />
(m) 3sinx−ln|x|+C (n) 5 2 x2 +3e −x +C (o) x−3e −x +C (p) 2 5 x5/2 − 16 5 x5/4<br />
<strong>Exercise</strong> 4.2<br />
1. Evaluate the integral by making the given substitution.<br />
∫<br />
(a) cos3xdx, u = 3x<br />
∫<br />
(b) x 2 (x 3 +2)dx, u = x 3 +2<br />
∫<br />
4<br />
(c) dx, u = 1+2x<br />
(1+2x)<br />
3<br />
∫ √ ( x+2)<br />
3<br />
(d) √ dx, u = √ x+2 x<br />
2. Evaluate the indefinite integral<br />
∫<br />
(a) 2x(x 2 +3) 4 dx<br />
∫ √x−1dx<br />
(c)<br />
∫<br />
(e)<br />
∫<br />
(g)<br />
dx<br />
5−3x<br />
1+4x<br />
√ dx<br />
1+x+2x<br />
2<br />
∫<br />
(i) cos2xdx<br />
∫<br />
(k) xsin(x 2 )dx<br />
∫<br />
(m) cos 4 xsinxdx<br />
∫<br />
(o) secxtanx √ 1+secxdx<br />
∫<br />
(b) (2x+1)(x 2 +x) 3 dx<br />
∫<br />
x 2<br />
(d) √<br />
x3 −2 dx<br />
∫<br />
(f)<br />
∫<br />
(h)<br />
2x+1<br />
x 2 +x−1 dx<br />
1<br />
√ √ dx x( x+1)<br />
2<br />
∫<br />
(j) cosx √ sinx+1dx<br />
∫<br />
sinx<br />
(l) √ dx cosx<br />
∫<br />
(n) sinx(cosx+3) 3/4 dx<br />
∫<br />
(p) cosxe sinx dx
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