e)

∫y′(s)y(s)2,ds\int \frac{y'(s)}{y(s)^2},ds Let (u = y(s)). Then (du = y’(s),ds). ∫u−2,du=−u−1+C=−1y(s)+C.\int u^{-2},du = -u^{-1}+C = -\frac{1}{y(s)}+C.

f)

∫v′(x)v(x),dx\int \frac{v'(x)}{\sqrt{v(x)}},dx Let (u=v(x)). ∫u−1/2,du=−2u1/2+C=2v(x)+C.\int u^{-1/2},du = -2u^{1/2}+C = 2\sqrt{v(x)}+C.

g)

∫u′(x)u(x)7,dx\int \frac{u'(x)}{u(x)^7},dx Let (u=u(x)). ∫u−7,du=−16u6+C=−16u(x)6+C.\int u^{-7},du = -\frac{1}{6u^6}+C = -\frac{1}{6u(x)^6}+C.

h)

∫v′(t)1+v(t)2,dt\int \frac{v'(t)}{1+v(t)^2},dt Let (u=v(t)). ∫du1+u2=arctan(u)+C=arctan(v(t))+C.\int \frac{du}{1+u^2} = \arctan(u)+C = \arctan(v(t))+C.

i)

∫y′(t)ey(t),dt\int y'(t)e^{y(t)},dt Let (u=y(t)). ∫eu,du=eu+C=ey(t)+C.\int e^u,du = e^u + C = e^{y(t)}+C.

j)

∫u′(x)sin(u(x)),dx\int u'(x)\sin(u(x)),dx Let (u=u(x)). ∫sinu,du=−cosu+C=−cos(u(x))+C.\int \sin u,du = -\cos u + C = -\cos(u(x))+C.

k)

∫y′(x)1−y(x)2,dx\int \frac{y'(x)}{\sqrt{1-y(x)^2}},dx Let (u=y(x)). ∫du1−u2=arcsin(u)+C=arcsin(y(x))+C.\int \frac{du}{\sqrt{1-u^2}} = \arcsin(u) + C = \arcsin(y(x)) + C.

l)

∫y′(x)(1+tan2(y(x))),dx\int y'(x)(1+\tan^2(y(x))),dx Use (1+^2 u = ^2 u). Let (u=y(x)). ∫sec2u,du=tan(u)+C=tan(y(x))+C.\int \sec^2 u,du = \tan(u) + C = \tan(y(x)) + C.

If you’d like, I can rewrite them in a more compact list or show derivative checks.