420 Chapter 7 • The Riemann Integral
- Find each of the following (see Exercise 11):
d r
(a) dx}o J3t^2 +5dt(c) -d^10 (2t^3 + 4) dt
dx x(e) - Jl+t2 dt
d 13 x-2
dx 0( g) .!!:_ ro _.!!:!.__
dx }3x-l t+4d 1 x2+ 1 dt
(i) dx x+l t + 2- Use Theorem 7.6.13 to find
(a) lo7r^14 sec^4 x tan x dx
(b) -d 1x (sin 3t + cos 4t) dt
dx 7r/ 2( d) d~ 1
3
·7
sin 5t dt
x2
(f) .!!:_ r sin( t^3 ) dt
dx Jo(h) .!!:_ r4 ..ft dt
dx J2x2+ 1
x3
(j) dd 1 -Vl + t dt
X x2(b) lo7r^12 cos^5 dx14. Use integration by parts to find each of the following:(a) j~dx x (^2) -1
(c) 1
4
lnxdx
(e) 1
9
efi dx
(g) 1
2
sin(ln x) dx
(i) j tan-^1 x dx
(b) j x^5 ~dx
(d) 1
3
x^2 ln JX dx
(f) 1
4
JXefi dx
(h) j sin-^1 x dx
(j) j xtan-^1 x dx
- Fill in the details requested in the proof of Theorem 7.6.16.
- Prove the first mean value theorem for integrals (Theorem 7.6.17). [Hint:
Use Theorem 7.5.2 and the intermediate value theorem.]
17. Prove the following variant of the first mean value theorem for integrals:
If f is monotone on [a, b], then 3 c E (a, b) such that l: f = f(a)(c - a)+
f(b)(b-c). [Hint: Consider the function g(x) = f(a)(x -a)+ f(b)(b-x).
Show that l: f falls between g(a) and g(b).]