1. hw 9 (1) Let g : (−1, ∞) → R be a function such that
g(3x4+ 4x3+ 1) = ln(x + 2).
Find g0(8).
(2) In class, we define the number sin−1x, −1 ≤ x ≤ 1 to be the unique solution to the equation sin θ = x when −π
2 ≤ θ ≤ π 2.
The assignment to each x ∈ [−1, 1] the solution of the above equation defines a function sin−1: [−1, 1] → [−π
2,π 2].
In this exercise, we try to compute the derivative of sin−1x with respect to x when −1 <
x < 1. At first, we observe that
sin(sin−1x) = x, x ∈ (−1, 1).
(a) Using chain rule to differentiate both side of the above equation, we can show that cos(sin−1x) · d
dxsin−1x = 1.
Express cos(sin−1x) in terms of x using cos2θ + sin2θ = 1 to find the derivative of sin−1x.
(b) Define a function f : [−1, 1] → R by f (x) =
Z x
−1
√ dt 1 − t2.
By fundamental Theorem of calculus I, we know that f0(x) = 1
√1 − x2, i.e. f is an antiderivative of 1
√
1 − x2. Use the change of variable formula t = sin θ to find f (x).
(3) For any x ∈ R, the equation tan θ = x has a unique solution when −π
2 < θ < π 2. The solution to this equation is denoted by tan−1x.
(a) Find d
dxtan−1x using the relation tan(tan−1x) = x.
(b) Define a function f : (−∞, ∞) → R by f (x) =
Z x 0
dt
1 + t2, x ∈ R.
Use the change of variable formula t = tan θ to find f (x).
(c) Evaluate the indefinite integral
Z dx
x2+ x + 1.
(d) Let a, b, c ∈ R such that b2− 4ac < 0. Evaluate the indefinite integral
Z dx
ax2+ bx + c.
(4) Let b1, · · · , bn and b01, · · · , b0n be real numbers. Define functions f, g : R → R by f (x) =
n
X
k=1
bksin(kx), g(x) =
n
X
k=1
b0ksin(kx).
(a) Evaluate Z π
−π
f (x)2dx. Express the answers in terms of b1, · · · , bk.
1
2
(b) Evaluate Z π
−π
f (x)g(x)dx. Express the answers in terms of b1, · · · , bk, and b01, · · · , b0k. (5) Let a1, · · · , an be distinct real numbers. Define a polynomial P (x) by
P (x) = (x − a1)(x − a2) · · · (x − an).
(a) Let Q be any real polynomial degree < n. Find Ai such that Q(x)
P (x) =
n
X
i=1
Ai
x − ai. Express Ai in terms of Q(x), P (x) and a1, · · · , an. (b) Use the definition
ln x = Z x
1
dt t to find the indefinite integral
Z Q(x) P (x)dx.
(c) Evaluate
Z x2+ x + 1
(x − 1)(x − 2)(x − 3)dx.
(6) For each natural number n, we define In=
Z π2
0
sinnxdx.
Use sinnx = sinn−1x sin x and the integration by part formula to find the relation between In and In−2.
(7) Evaluate
Z 2π 0
dx
(2 + cos x)(3 + cos x). Using the following steps.
(a) Find A and B such that 1
(2 + cos x)(3 + cos x) = A
2 + cos x+ B 3 + cos x. (b) Evaluate
Z 2π 0
dx a + cos x
where a = 2, 3. To do this you use the change of variable t = tanx2 and show that cos x =1 − t2
1 + t2, sin x = 2t 1 + t2.
