MATH1014 Calculus II Problem Set 4 L01 (Spring 2025)C/C++

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Calculus II

Problem Set 4

L01 (Spring 2025)

1.     For each of the following rational functions  f, evaluate the antiderivative  ∫ f(x)dx.

2.     Evaluate the following antiderivatives.

3.     (a)     Using the factorization  x4  + 1 =  (x2  − √2x + 1)(x2  + √2x + 1), evaluate

(b)    Using (a) and the substitution  u  = √tan x, evaluate

4.     Evaluate the antiderivative

 

using the substitution 

5.     (a)     Show that the polynomial  x 3  + 3x + 1   has exactly one real root.

(b)    Let  r   be the real root of  x 3  + 3x + 1.     Using a partial fraction decomposition, evaluate

  dai 写MATH1014 Calculus II Problem Set 4 L01 (Spring 2025)C/C++

in terms of  r.

6.     Evaluate the antiderivatives of each of the following trigonometric rational functions  f.

7.     Let  a  be apositive real number.     Evaluate

 

for each of the following cases:

(a)    0 < a < 1,

(Try to find antiderivatives just on   (−π, π); antiderivativeson  ℝ  are too complicated.)

(b)   a = 1,

(c)    a > 1.

8.     Evaluate the antiderivative

 

using the substitution  t = tan(x/2).

9.     Evaluate each of the following improper integrals if it converges.

 

Hint:         In (c), break the interval into two halves, and let    in the second half.

10.   Let  f: [0, +∞) → ℝ   be the function

 

(a)    Find  f ′ (x)  for every  x  ∈ (0, +∞)