3.7 Definite integrals

1.

(a) $-f(x)$(b) $f(x+1) - f(x)$(c) $${2x \int_{0}^{2x}f(t)dt + 2x^2 f(2x)}$$

2.

(a) $${\frac{32}{3}}$$(b) $${\frac{1}{4}}$$(c) $1$(d) $${\frac{1}{2} - \frac{1}{2 e}}$$(e) $${\log{(\frac{3}{2})}}$$

3. omitted

4.

(a) $${\frac{\pi}{4} = \int_{0}^{1} \frac{dx}{1+x^2} \leq \int_{0}^{1} \frac{dx}{1 + x^{n}} < 1}$$

(b)

$$\frac{1}{2(n+1)} < \int_{0}^{1}\frac{x^{n}}{1+x} dx < \frac{1}{n+1} < \frac{1}{n}$$

5.

(a) $${\log{2}}$$(b) $${\log{(1 + \sqrt{2})}}$$(c) $${\frac{1}{3}}$$