Higher Engineering Mathematics, Sixth Edition
422 Higher Engineering Mathematics The integral, ∫ xcosxdx, is not a ‘standard integral’ and it can only be determined by using ...
Integration by parts 423 Problem 8. Evaluate ∫ 9 1 √ xlnxdx, correct to 3 significant figures. Letu=lnx, from which du= dx x and ...
424 Higher Engineering Mathematics i.e. ( 1 + a^2 b^2 )∫ eaxcosbxdx = 1 b eaxsinbx+ a b^2 eaxcosbx i.e. ( b^2 +a^2 b^2 )∫ eaxcos ...
Integration by parts 425 In determining a Fourier series to repre- sentf(x)=xin the range−πtoπ, Fourier coefficients are given ...
Chapter 44 Reduction formulae 44.1 Introduction When using integration by parts in Chapter 43, an integral such as ∫ x^2 exdx re ...
Reduction formulae 427 From equation (1),In=xnex−nIn− 1 Hence ∫ x^3 exdx=I 3 =x^3 ex− 3 I 2 I 2 =x^2 ex− 2 I 1 I 1 =x^1 ex− 1 I ...
428 Higher Engineering Mathematics From equation (2), ∫ t^3 costdt=I 3 =t^3 sint+ 3 t^2 cost− 3 ( 2 )I 1 and I 1 =t^1 sint+ 1 t^ ...
Reduction formulae 429 Problem 6.∫ Use a reduction formula to determine x^3 sinxdx. Using equation (3), ∫ x^3 sinxdx=I 3 =−x^3 c ...
430 Higher Engineering Mathematics du dx =(n− 1 )sinn−^2 xcosx and du=(n− 1 )sinn−^2 xcosxdx and let dv=sinxdx, from which, v= ∫ ...
Reduction formulae 431 =4[(− 0. 054178 − 0. 1020196 − 0. 2881612 )−(− 0. 533333 )] = 4 ( 0. 0889745 )= 0. 356 Problem 10. Determ ...
432 Higher Engineering Mathematics Hence ∫ cos^4 xdx = 1 4 cos^3 xsinx+ 3 4 ( 1 2 cosxsinx+ 1 2 x ) = 1 4 cos^3 xsinx+ 3 8 cosxs ...
Reduction formulae 433 = ∫ tann−^2 xsec^2 xdx−In− 2 i.e.In= tann−^1 x n− 1 −In− 2 Whenn=7, I 7 = ∫ tan^7 xdx= tan^6 x 6 −I 5 I 5 ...
434 Higher Engineering Mathematics =x(lnx)n−n ∫ (lnx)n−^1 dx i.e.In=x(lnx)n−nIn− 1 Whenn=3, ∫ (lnx)^3 dx=I 3 =x(lnx)^3 − 3 I 2 I ...
Chapter 45 Numerical integration 45.1 Introduction Even with advanced methods of integration there are many mathematical functio ...
436 Higher Engineering Mathematics Problem 1. (a) Use integration to evaluate, correct to 3 decimal places, ∫ 3 1 2 √ x dx(b) Us ...
Numerical integration 437 With 6 intervals, each will have a width of π 2 − 0 6 i.e. π 12 rad (or 15◦) and the ordinates occur a ...
438 Higher Engineering Mathematics Problem 4. Use the mid-ordinate rule with (a) 4 intervals, (b) 8 intervals, to evaluate ∫ 3 1 ...
Numerical integration 439 Now try the following exercise Exercise 174 Further problemson the mid-ordinate rule In Problems 1 to ...
440 Higher Engineering Mathematics y 1 y 2 y 3 y 4 y 2 n 1 a ddd b yf(x) x y O Figure 45.4 ≈ 1 3 d[(y 1 +y 2 n+ 1 )+ 4 (y 2 +y ...
Numerical integration 441 With 6 intervals, each will have a width of π 3 − 0 6 i.e. π 18 rad (or 10◦), and the ordinates will o ...
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