8.6. EXERCISES 203

To summarize, here is a short table of auspicious substitutions corresponding to certainexpressions.

Table Of Auspicious Substitutions

Expression a2 +b2x2 a2 −b2x2 a2x2 −b2

Trig. substitution bx = a tan(u) bx = asin(u) ax = bsec(u)

Hyperbolic substitution bx = asinh(u)

Of course there are no “magic bullets” but these substitutions will often simplify anexpression enough to allow you to find an antiderivative. These substitutions are oftenespecially useful when the expression is enclosed in a square root.

8.6 Exercises1. Find the antiderivatives.

(a)∫ x√

4−x2dx

(b)∫ 3√

36−25x2dx

(c)∫ 3√

16−25x2dx

(d)∫ 1√

4−9x2dx

(e)∫ 1√

36−x2dx

(f)∫ (√

9−16x2)3

dx

(g)∫ (√

16− x2)5

dx

(h)∫ √

25−36x2 dx

(i)∫ (√

4−9x2)3

dx

(j)∫ √

1−9x2 dx

2. Find the antiderivatives.

(a)∫ √

36x2 −25dx

(b)∫ √

x2 −4dx

(c)∫ (√

16x2 −9)3

dx

(d)∫ √

25x2 −16dx

3. Find the antiderivatives.

(a)∫ 1

26+x2−2x dx Hint: Complete thesquare.

(b)∫ √

x2 +9dx

(c)∫ √

4x2 +25dx

(d)∫

x√

4x4 +9dx

(e)∫

x3√

4x4 +9dx

(f) ∗ ∫ 1

(16+25(x−3)2)2 dx

(g)∫ 1

261+25x2−150x dx Hint: Completethe square.

(h)∫ (√

25x2 +9)3

dx

(i)∫ 1

25+16x2 dx

4. Find the antiderivatives. Hint: Complete the square.