Even More Integration Problems Hints
If you just want to check your answers, use Maple (be aware Maple may
present the answer in a superficially different form than you obtain; try
simplifying Maple’s answer MINUS your answer to see if you get zero).
1. Try a substitution u= 1 −x. Or integrate by parts with u=x, v0=
(1 −x)1/2.
2. Do a partial fraction decomposition.
3. Integrate by parts with u=t, v0=e2t.
4. Substitute z= 3 sin(u), dz = 3 cos(u)du. The integral becomes 9 Zcos2(u)du.
Now use the identity cos2(u) = 1/2 + cos(2u)/4.
5. Substitute x= sin(u) (so u= arcsin(x)), dx = cos(u)du.
6. Substitute u= 9 −z2, so du =−2z dz, or z dz =−du/2.
Harder Problem Hints
1. Split the integrand into exln(x) and ex/x. Then integrate exln(x) by
parts with u= ln(x), dv =exdx.
2. Many approaches. Integrate by parts with u= arcsin(x)2, dv =dx,
then do the substitution x= sin(u). Finish with another obvious inte-
gration by parts.
3. Note that x4+ 4 = (x2
−2x+ 2)(x2+ 2x+ 2), then do a partial fraction
decomposition.
4. Integrate by parts with u= ln(√x+√1 + x), dv =dx. Do the remain-
ing integral with the substitution u=√x+ 1.
5. Substitute z=√y, then integrate by parts with u= arcsin(z). Finish
the remaining integral with the substitution z= sin(t) and use the
identity sin2(t) = 1/2−cos(2t)/2.
6. Use the substitution u= 1 + √x+ 1, followed by the substitution
q=q1 + √u(you could amalgamate the substitutions, but it’s alge-
braically less error-prone to do it in two steps.) By the way, Maple
won’t touch this integral!
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