3.7.1 Integration Short Questions (Question 1)

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

Solution:
(a)
$\begin{array}{l}\int \left(\frac{3}{{x}^{2}}-\frac{5}{2{x}^{3}}+2\right)dx\\ =\int \left(3{x}^{-2}-\frac{5{x}^{-3}}{2}+2\right)dx\\ =-3{x}^{-1}+\frac{5{x}^{-2}}{4}+2x+c\\ =-\frac{3}{x}+\frac{5}{4{x}^{2}}+2x+c\end{array}$

(b)
$\begin{array}{l}\int {x}^{2}\left({x}^{5}+2x\right)dx\\ =\int \left({x}^{7}+2{x}^{3}\right)\text{}dx\\ =\frac{{x}^{8}}{8}+\frac{2{x}^{4}}{4}+c\\ =\frac{{x}^{8}}{8}+\frac{{x}^{4}}{2}+c\end{array}$

(c)
$\begin{array}{l}\int \frac{3{x}^{4}+2x}{{x}^{3}}dx\\ =\int \left(\frac{3{x}^{4}}{{x}^{3}}+\frac{2x}{{x}^{3}}\right)\text{}dx=\int \left(3x+2{x}^{-2}\right)\text{}dx\\ =\frac{3{x}^{2}}{2}-\frac{2}{x}+c\end{array}$

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

(e)
$\begin{array}{l}\int {\left(5x-1\right)}^{3}dx\\ =\frac{{\left(5x-1\right)}^{4}}{\left(4\right)\left(5\right)}+c\\ =\frac{1}{20}{\left(5x-1\right)}^{4}+c\end{array}$

(f)
$\begin{array}{l}\int \frac{3}{{\left(4x+7\right)}^{8}}dx=\int 3{\left(4x+7\right)}^{-8}\text{}dx\\ =\frac{3{\left(4x+7\right)}^{-7}}{\left(-7\right)\left(4\right)}+c\\ =-\frac{3}{28{\left(4x+7\right)}^{7}}+c\end{array}$