Page 134 - Calculus Demystified

The Integral
CHAPTER 4
2. Calculate each of the following indefinite integrals: 121

2
(a) x sin x dx
2
(b) (3/x) ln x dx

(c) sin x · cos xdx

(d) tan x · ln cos xdx

2
(e) sec x · e tan x dx
2 43
(f) (2x + 1) · (x + x + 7) dx
3. Use Riemann sums to calculate each of the following integrals:
2 2
(a) x + xdx
1
1 2
(b) (−x /3)dx
−1
4. Use the FundamentalTheorem of Calculus to evaluate each of the following
integrals:
3
3
2
(a) x − 4x + 7 dx
1
6 2
(b) xe x − sin x cos xdx
2
4
2
(c) (ln x/x) + x sin x dx
1
2
3
2
(d) 1 tan x − x cos x dx
e
2
(e) (ln x /x) dx
1
8
3
2
3
(f) x · cos x sin x dx
4
5. Calculate the area under the given function and above the x-axis over the
indicated interval.
2
(a) f(x) = x + x + 6 [2, 5]
(b) g(x) = sin x cos x [0,π/4]
(c) h(x) = xe x 2 [1, 2]
(d) k(x) = ln x/x [1,e]
6. Draw a careful sketch of each function on the given interval, indicating
subintervals where area is positive and area is negative.
3
(a) f(x) = x + 3x [−2, 2]
(b) g(x) = sin 3x cos 3x [−2π, 2π]
(c) h(x) = ln x/x [1/2,e]
3 x
(d) m(x) = x e 4 [−3, 3]