194 CHAPTER 8. METHODS FOR FINDING ANTIDERIVATIVES

(a)∫

sec(3x) dx(b)

∫sec2 (3x) tan(3x) dx

(c)∫ 1

3+5x2 dx

(d)∫ 1√

5−4x2dx

(e)∫ 3

x√

4x2−5dx

7. Find the indicated antiderivatives.

(a)∫

xcosh(x2 +1

)dx

(b)∫

x35x4dx

(c)∫

sin(x)7cos(x) dx

(d)∫

xsin(x2)

dx

(e)∫

x5√

2x2 +1dx Hint: Let u =2x2 +1.

8. Find∫

sin2 (x) dx. Hint: Derive and use sin2 (x) = 1−cos(2x)2 .

9. Find the indicated antiderivatives.

(a)∫ lnx

x dx

(b)∫ x3

3+x4 dx

(c)∫ 1

x2+2x+2 dx Hint: Complete thesquare in the denominator and thenlet u = x+1. Remember the arctanfunction.

(d)∫ 1√

4−x2dx

(e)∫ 1

x√

x2−9dx Hint: Let x = 3u.

(f)∫ ln(x2)

x dx

(g) Find∫ x3√

6x2+5dx

(h) Find∫

x 3√

6x+4dx

10. Find the indicated antiderivatives.

(a)∫

x√

2x+4dx(b)

∫x√

3x+2dx(c)

∫ 1√36−25x2

dx

(d)∫ 1√

9−4x2dx

(e)∫ 1√

1+4x2dx

(f)∫ x√

(3x−1)dx

(g)∫ x√

5x+1dx

(h)∫ 1

x√

9x2−4dx

(i)∫ 1√

9+4x2dx

11. Find∫ 1

x1/3+x1/2 dx. Hint: Try letting x = u6 and use long division.

12. Suppose f is a function defined on R and it satisfies the functional equation given byf (a+b) = f (a)+ f (b) . Suppose also f ′ (0) = k. Find f (x) .

13. Suppose f is a function defined on R having values in (0,∞) and it satisfies thefunctional equation f (a+b) = f (a) f (b) . Suppose also f ′ (0) = k. Find f (x) .

14. Suppose f is a function defined on (0,∞) having values in R and it satisfies thefunctional equation f (ab) = f (a)+ f (b) . Suppose also f ′ (1) = k. Find f (x) .

15. Suppose f is a function defined on R and it satisfies the functional equation

f (a+b) = f (a)+ f (b)+3ab.

Suppose also that limh→0f (h)

h = 7. Find f (x) if possible.