Integration by substitution - Mathcentre

Writing f(u) = sin uand g(x) = √ xthen g ′ (x) = 1 2 x−1/2 = 12x 1/2 = 12 √ x .Further, f(g(x)) = sin √ x.Hencewewritethegivenintegralas∫2sin √ x 12 √ x dxwhichisoftheform∫2f(g(x))g ′ (x) dxwith fand gasgivenabove.Asbeforethesubstitution u = g(x) = √ xproducestheintegral∫ ∫2 f(u) du = 2 sin u dufromwhich∫2sin u du = −2 cosu + c= −2 cos √ x + cExercises21.Ineachcasetheintegrandcanbewrittenas f(g(x)) g ′ (x).Identifythefunctions fand gandusethegeneralresultonpage7tocompletetheintegration.∫∫∫(a) 2xe x2−5 dx (b) −2x sin(1 − x 2 cos x)dx (c)1 + sin x dx.2.Ineachcaseusethegivensubstitutiontofindtheintegral:∫(a) −2xe −x2 dx, u = −x 2 .∫(b) x sin(2x 2 )dx, u = 2x 2 .(c)∫ 50x 3√ x 4 + 1dx, u = x 4 + 1.3.Ineachcaseuseasuitablesubstitutiontofindtheintegral.∫(a) 5x √ ∫∫dx1 − x 2 dx (b) √ √ (c) x 4 (1 + x 5 ) 3 dxx(1 + x)2∫x 3∫∫(d) √x4 + 16 dx (e) cosx1x 3(f) √(5 + sin x) 2 0 x4 + 12 dx∫(g) 5x 2√ ∫∫1 − x 3 dx (h) e cos x sin x dx (i) e sin x cosxdx.www.mathcentre.ac.uk 9 c○mathcentre2009