Quiz 3B for MATH 105 SECTION 205
February 11, 2015
Family Name
Given Name
Student Number
1. (a) (0.5 points) Is it true that any continuous function has an antiderivative?(just put ‘Yes’ or ‘No’)
(a)
(b) (0.5 points) Find an antiderivative of |x|.
(b)
1
x
(c) (0.5 points) Compute lim
x→0
Z
x2
cos(t2 ) dt.
x
(c)
Z r q
√
(d) (0.5 points) Evaluate
x x x dx.
(d)
Z
(e) (0.5 points) Evaluate
10x · 32x dx. (Hint: ax = ex ln a ).
(e)
Z
(f) (0.5 points) Evaluate
cos(2x)
dx (Hint: cos(2x) = 2 cos2 (x) − 1 and sin2 (x) + cos2 (x) = 1).
cos(x) − sin(x)
(f)
Z
(g) (0.5 points) Evaluate
cos(2x)
dx.
· sin2 (x)
cos2 (x)
(g)
Z
r
(h) (0.5 points) Evaluate
1+x
+
1−x
r
1−x
1+x
!
dx (Hint: Simplify the integrand).
(h)
Z
(i) (0.5 points) Evaluate
√
√
dx
√
(Hint: Let u = 6 x).
3
x+ x
(i)
Z
(j) (0.5 points) Evaluate
tan−1 (x) dx.
(j)
Z 1
(k) (0.5 points) Evaluate
ln(ln x) +
dx.
ln x
(k)
(l) (0.5 points) Find the area of the region bounded by the graph of ln x and x-axis between x = e−1 and
x = e.
(l)
(m) (0.5 points) Compute lim
n→∞
1
3
3
1
+
2
+
·
·
·
+
n
.
n4
(n) (0.5 points) Compute lim n
n→∞
(m)
1
1
1
.
+
+
·
·
·
+
n2 + 1 n2 + 2 2
2n2
(n)
1
π
2π
n−1
(o) (0.5 points) Compute lim
sin + sin
+ · · · + sin
π .
n→∞ n
n
n
n
(o)
Z
(p) (0.5 points) Evaluate
sin3 (x) cos2 (x) dx.
(p)
Z
(q) (0.5 points) Evaluate
tan(x) sec2 (x) dx.
(q)
Z
(r) (0.5 points) Evaluate
tan2 (x) sec3 (x) dx.
(r)
Z
2. (1 point) For any m, n, let I(m, n) =
cosm (x) sinn (x) dx, then if m + n 6= 0, show that
cosm−1 (x) sinn+1 (x) m − 1
+
I(m − 2, n)
m+n
m+n
cosm+1 (x) sinn−1 (x)
n−1
= −
+
I(m, n − 2).
m+n
m+n
I(m, n) =
Z
3. (1 point) Show that
sinn−1 (x) cos(x) n − 1
+
sin (x) dx = −
n
n
n
Z
sinn−2 (x) dx.
Z
4. (1 point) Use the integration by parts to get a reduction formula of the integral
ex sinn (x) dx.
Your Score:
/12