MATH 296 PROBLEMS 11

IGOR KRIZ

Regular problems:

1. Compute the following integrals:

(a) ∫ x arctan(x)dx

(b) ∫ \frac{e^{x}}{1+e^{2x}} dx

(c) ∫ x^{2}\sqrt{x + 1}dx.

2. Using integration of rational functions, compute:

(a) ∫ \frac{2x^{2}+5x−1}{x^{3}+x^{2}−2x}dx

(b) ∫ \frac{3x^{2}+2x−2}{x^{3}−1}dx.

3. Suppose f : ℝ^{1} → ℝ^{3} is such that ||f(t)|| = 1 (the Euclidean norm || ( \begin{matrix}x \\ y \\ z \end{matrix} ) || =

\sqrt{x^{2} + y^{2} + z^{2}}) for every t. Prove that Df_{t}.f(t) = 0 for every t ∈ ℝ.

4. Define for f : ℝ^{3} → ℝ, ∇^{2}f = \frac{∂^{2}f}{∂x^{2}} + \frac{∂^{2}f}{∂y^{2}} + \frac{∂^{2}f}{∂z^{2}} . Similarly, for f : ℝ^{3} → ℝ^{3},

f(x) = ( \begin{matrix}f_{1}(x) \\ f_{2}(x) \\ f_{3}(x) \end{matrix} ), let ∇^{2}f = ( \begin{matrix}∇^{2}f_{1} \\ ∇^{2}f_{2} \\ ∇^{2}f_{3} \end{matrix} ). (This is called the Laplacian.) Prove that

curl(curl(f)) = grad(div(f)) − ∇^{2}f.

Challenge problems:

5. Integrate:

(a) ∫ \frac{xdx}{4−x^{2}+\sqrt{4−x^{2}}}

(b) ∫ \frac{dx}{2 sin x−cos x+5}

(c) ∫ _{\frac{dx}{x+\sqrt{\frac{x+2}{x+1}}}} .

6. Put f(0, 0) = 0, and

f(x, y) = \frac{xy(x^{2} − y^{2})}{x^{2} + y^{2}}

if (x, y) ≠ (0, 0). Prove that (a) f, \frac{∂f}{∂x}, \frac{∂f}{∂y} are continuous in ℝ^{2};

(b) \frac{∂^{2}f}{∂x∂y} and \frac{∂^{2}f}{∂y∂x} exist at every point of ℝ^{2} and are continuous except at (0, 0);

(c) \frac{∂^{2}f}{∂x∂y} (0, 0) = 1 \frac{∂^{2}f}{∂y∂x}(0, 0) = −1