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Integration Problems Hints: Techniques for Solving Integrals, Assignments of Calculus

Hints and techniques for solving various integration problems. It covers methods such as substitution, partial fraction decomposition, integration by parts, and the use of identities. The document also includes examples of harder integration problems and their respective solutions. This resource is ideal for students and researchers in mathematics and related fields who need assistance in solving complex integrals.

Typology: Assignments

Pre 2010

Uploaded on 08/19/2009

koofers-user-ra0
koofers-user-ra0 🇺🇸

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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) = 1/2cos(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!
1

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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, v′^ = (1 − x)^1 /^2.
  2. Do a partial fraction decomposition.
  3. Integrate by parts with u = t, v′^ = e^2 t.
  4. Substitute z = 3 sin(u), dz = 3 cos(u) du. The integral becomes 9

∫ cos^2 (u) du. Now use the identity cos^2 (u) = 1/2 + cos(2u)/4.

  1. Substitute x = sin(u) (so u = arcsin(x)), dx = cos(u) du.
  2. Substitute u = 9 − z^2 , so du = − 2 z dz, or z dz = −du/2.

Harder Problem Hints

  1. Split the integrand into ex^ ln(x) and ex/x. Then integrate ex^ ln(x) by parts with u = ln(x), dv = ex^ dx.
  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 x^4 + 4 = (x^2 − 2 x + 2)(x^2 + 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.

  1. 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 sin^2 (t) = 1/ 2 − cos(2t)/2.

  1. Use the substitution u = 1 +

x + 1, followed by the substitution q =

√ 1 +

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!