Solutions - 0130-04

(a)

(1) Definition of improper integral:

$${\int }_1^{\infty }\frac{dx}{x^{19/20}}=\underset{R\to \infty }{\lim }{\int }_1^R\frac{dx}{x^{19/20}}$$

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(2) Antiderivative and limit:

$$\begin{array}{c}\gg \gg \underset{R\to \infty }{\lim }20x^{1/20}{|}_1^R\\ \\ \gg \gg \underset{R\to \infty }{\lim }20(R^{1/20}-1)\gg \gg +\infty \text{(diverges)}\end{array}$$

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(b)

(1) Definition of improper integral:

$${\int }_0^5\frac{dx}{x^{20/19}}=\underset{R\to 0^{+}}{\lim }{\int }_R^5\frac{dx}{x^{20/19}}$$

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(2) Antiderivative and limit:

$$\begin{array}{c}\gg \gg \underset{R\to 0^{+}}{\lim }-19x^{-1/19}{|}_R^5\\ \\ \gg \gg \underset{R\to 0^{+}}{\lim }-19(5^{-1/19}-R^{-1/19})\gg \gg +\infty \text{(diverges)}\end{array}$$

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(c)

(1) Definition of improper integral:

$${\int }_1^{\infty }{e}^{-x}dx=\underset{R\to \infty }{\lim }{\int }_1^R{e}^{-x}dx$$

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(2) Antiderivative and limit:

$$\begin{array}{c}\gg \gg \underset{R\to \infty }{\lim }-e^{-x}{|}_1^R\\ \\ \gg \gg \underset{R\to \infty }{\lim }-e^{-R}--e^{-1}\gg \gg e^{-1}\text{(converges)}\end{array}$$