3.7.1 Integration Short Questions (Question 1)

Question 1:

\begin{array}{l} Find the integral of each of the following . \\ (a) \int (\frac{3}{x^{2}} - \frac{5}{2 x^{3}} + 2) d x \\ (b) \int x^{2} (x^{5} + 2 x) d x \\ (c) \int \frac{3 x^{4} + 2 x}{x^{3}} d x \\ (d) \int \frac{(7 + x) (7 - x)}{x^{4}} d x \\ (e) \int {(5 x - 1)}^{3} d x \\ (f) \int \frac{3}{{(4 x + 7)}^{8}} d x \end{array}

Solution:
(a)

\begin{array}{l} \int (\frac{3}{x^{2}} - \frac{5}{2 x^{3}} + 2) d x \\ = \int (3 x^{- 2} - \frac{5 x^{- 3}}{2} + 2) d x \\ = - 3 x^{- 1} + \frac{5 x^{- 2}}{4} + 2 x + c \\ = - \frac{3}{x} + \frac{5}{4 x^{2}} + 2 x + c \end{array}

(b)

\begin{array}{l} \int x^{2} (x^{5} + 2 x) d x \\ = \int (x^{7} + 2 x^{3}) d x \\ = \frac{x^{8}}{8} + \frac{2 x^{4}}{4} + c \\ = \frac{x^{8}}{8} + \frac{x^{4}}{2} + c \end{array}

(c)

\begin{array}{l} \int \frac{3 x^{4} + 2 x}{x^{3}} d x \\ = \int (\frac{3 x^{4}}{x^{3}} + \frac{2 x}{x^{3}}) d x = \int (3 x + 2 x^{- 2}) d x \\ = \frac{3 x^{2}}{2} - \frac{2}{x} + c \end{array}

(d)

\begin{array}{l} \int \frac{(7 + x) (7 - x)}{x^{4}} d x = \int (\frac{49 - x^{2}}{x^{4}}) d x \\ = \int (\frac{49}{x^{4}} - \frac{1}{x^{2}}) d x = \int (49 x^{- 4} - x^{- 2}) d x \\ = \frac{49 x^{- 3}}{- 3} + \frac{1}{x} + c \\ = - \frac{49}{3 x^{3}} + \frac{1}{x} + c \end{array}

(e)

\begin{array}{l} \int {(5 x - 1)}^{3} d x \\ = \frac{{(5 x - 1)}^{4}}{(4) (5)} + c \\ = \frac{1}{20} {(5 x - 1)}^{4} + c \end{array}

(f)

\begin{array}{l} \int \frac{3}{{(4 x + 7)}^{8}} d x = \int 3 {(4 x + 7)}^{- 8} d x \\ = \frac{3 {(4 x + 7)}^{- 7}}{(- 7) (4)} + c \\ = - \frac{3}{28 {(4 x + 7)}^{7}} + c \end{array}