Mathematical Methods for Physics and Engineering - Matematica.NET

2.2 INTEGRATIONfound near the end of subsection 2.1.1. A few are presented below, using the formgiven in (2.30):∫∫adx= ax + c, ax n dx = axn+1n +1 + c,∫∫e ax dx = eaxaa + c, dx = a ln x + c,x∫∫a sin bx−a cos bxa cos bx dx = + c, a sin bx dx = + c,bb∫∫a tan bx dx =−a ln(cos bx)b+ c,a(a 2 + x 2 dx x)=tan−1 + c,a∫−1( √a2 − x dx x)2=cos−1 + c,a∫∫a cos bx sin n bx dx = a sinn+1 bxb(n +1)a sin bx cos n bx dx = −a cosn+1 bxb(n +1)∫+ c,+ c,1( √a2 − x dx x)2=sin−1 + c,awhere the integrals that depend on n are valid for all n ≠ −1 andwherea and bare constants. In the two final results |x| ≤a.2.2.4 Integration of sinusoidal functionsIntegrals of the type ∫ sin n xdx and ∫ cos n xdx may be found by using trigonometricexpansions. Two methods are applicable, one for odd n and the other foreven n. They are best illustrated by example.◮Evaluate the integral I = ∫ sin 5 xdx.Rewriting the integral as a product of sin x and an even power of sin x, and then usingthe relation sin 2 x =1− cos 2 x yields∫I = sin 4 x sin xdx∫= (1 − cos 2 x) 2 sin xdx∫= (1 − 2cos 2 x +cos 4 x)sinxdx∫= (sin x − 2sinx cos 2 x +sinx cos 4 x) dx= − cos x + 2 3 cos3 x − 1 5 cos5 x + c,where the integration has been carried out using the results of subsection 2.2.3. ◭63