Calculus: Analytic Geometry and Calculus, with Vectors

4.1 Indefinite integrals 211

Let x be confined to an interval I over which two given functions u and v are
differentiable. The standard formula

(1) d u(x)v(x) = u(x)v'(x) + v(x)u'(x)

then (why?) gives the formula

(2) f[u(x)v'(x) + v(x)u'(x)] dx = u(x)v(x) + c.

In case the separate integrals are cooperative enough to exist, we can (why?)

put (2) in the form

(3) fu(x)v'(x) dx + fv(x)u'(x) dx = u(x)v(x) + c

and transpose to obtain the formula

(4) fu(x)v'(x) dx = u(x)v(x) - fv(x)u'(x) dx,

which is known as the formula for integration by parts. For the particular case

in which u(x) = x and v(x) = ex, show that (4) reduces to

(5) fxex dx = xex - ex + c.

Finally, check (5) by showing that the derivative of the right side actually is the

integrand.

15 Read and work the preceding problem again.

16 With Problem 14 out of sight, start with the formula for the derivative

of a product and construct the formula for integration by parts and give an

application of it.

17 Start with the function fo for which fo(x) = 1 when -1 < x < 1 and

determine the natures of the functions f1, f2, fs, - for which the formulas

fn(x) = f.-1(x), fn(x) = ff.-,(x) dx, (n = 1,2,3,. ..)

are valid. .ns.: There exist constants c1, c2, C3, such that

fi(x) =x+c1

f2(x) = $x2 + clx + C2

f s(x) = -19X' + VCix.2 + c2x + C8

f4(x) = VITX4 + $Clxs + c2x2 + C3X + Ca

etcetera. Remark: These things will appear later.

18 Prove that the first of the formulas

f sgn x dx = jxj + c1, f sgn x dx = Jxl + C2

is correct when x > 0 and the second is correct when x < 0. Prove that there

is no constant c such that the formula

f sgn x dx = lxI + C

is correct when x = 0.

19 This problem contains hundreds of parts, and there is much to be saidfor

spending several hours or days solving a considerable number of them. To