Standard integration 373
=⎡
⎣θ3
2
3
2+2 θ1
2
1
2⎤
⎦41=[
2
3√
θ^3 + 4√
θ] 41={
2
3√
( 4 )^3 + 4√
4}
−{
2
3√
( 1 )^3 + 4√
( 1 )}={
16
3+ 8}
−{
2
3+ 4}= 51
3+ 8 −2
3− 4 = 82
3Problem 14. Evaluate∫ π
2
03sin2xdx.∫ π
2
03sin2xdx=[
( 3 )(
−1
2)
cos2x]π 20=[
−3
2cos2x]π 20={
−3
2cos2(π2)}
−{
−3
2cos2( 0 )}={
−3
2cosπ}
−{
−3
2cos0}={
−3
2(− 1 )}
−{
−3
2( 1 )}
=3
2+3
2= 3Problem 15. Evaluate∫ 214cos3tdt.∫ 2
14cos3tdt=[
( 4 )(
1
3)
sin3t] 21=[
4
3sin3t] 21={
4
3sin6}
−{
4
3sin3}Note that limits of trigonometric functions are always
expressed in radians—thus, for example, sin6 means
the sine of 6radians=− 0. 279415 ...
Hence
∫ 214cos3tdt={
4
3(− 0. 279415 ...)}
−{
4
3( 0. 141120 ...)}=(− 0. 37255 )−( 0. 18816 )=− 0. 5607Problem 16. Evaluate(a)∫ 214e^2 xdx (b)∫ 413
4 udu,each correct to 4 significant figures.(a)∫ 214e^2 xdx=[
4
2e^2 x] 21=2[e^2 x]^21 =2[e^4 −e^2 ]=2[54. 5982 − 7 .3891]= 94. 42(b)∫ 413
4 udu=[
3
4lnu] 41=3
4[ln4−ln1]=3
4[1. 3863 −0]= 1. 040Now try the following exerciseExercise 146 Further problems on definite
integralsIn problems 1 to 8, evaluate the definite integrals
(where necessary, correct to 4 significant figures).- (a)
∫ 415 x^2 dx (b)∫ 1− 1−3
4t^2 dt
[
(a) 105 (b)−1
2]- (a)
∫ 2− 1( 3 −x^2 )dx (b)∫ 31(x^2 − 4 x+ 3 )dx[
(a) 6 (b)− 11
3]- (a)
∫π03
2cosθdθ (b)∫ π
2
04cosθdθ[(a) 0 (b) 4]- (a)
∫ π
3
π
62sin2θdθ (b)∫ 203sintdt[(a) 1 (b) 4.248]- (a)
∫ 105cos3xdx (b)∫ π 603sec^22 xdx[(a) 0.2352 (b) 2.598]