Definite integral

Exercise

1.

For $f(t)$ is continuous， find $g(x)$．

(a)

$${g(x) = \frac{d}{dx}\int_{x}^{b}f(t)dt}$$

(b)

$${g(x) = \frac{d}{dx}\int_{x}^{x+1}f(t)dt}$$

(c)

$${g(x) = \frac{d}{dx}\int_{0}^{2x}x^{2}f(t)dt}$$

2.

Evaluate the following definite integral．

(a)

$${\int_{1}^{5}2\sqrt{x-1}dx}$$

(b)

$${\int_{1}^{2}\frac{2-t}{t^{3}}dt}$$

(c)

$${\int_{0}^{\frac{\pi}{2}}\cos{x}dx}$$

(d)

$${\int_{0}^{1}xe^{-x^{2}}dx}$$

(e)

$${\int_{0}^{\log{2}} \frac{e^{x}}{e^{x} + 1}dx}$$

4.

Prove the following inequality．

(a)

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

(b)

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

5.

Find the following limit.．

(a)

$${\lim_{n \rightarrow \infty} \left(\frac{1}{n+1} + \frac{1}{n + 2} + \cdots + \frac{1}{2n} \right)}$$

(b)

$${\lim_{n \rightarrow \infty} \sum_{i = 1}^{n} \sqrt{\frac{1}{n^2 + i^2}}}$$

(c)

$${\lim_{x \rightarrow 0} \int_{0}^{x} \tan{(t^2)}dt}$$