Question

Find the following integrals:
i) $\int \frac{x^3-1}{x^2}dx$
ii) $\int (x^{\frac{2}{3}}+1)dx$
iii) $\int (x^{\frac{3}{2}}+2e^x-\frac{1}{x})dx$

Answer 1

(i) $\int \frac{x^3-1}{x^2}dx$

$=\int \left(x-x^{-2}\right)dx$

$=\frac{x^2}{2}-\frac{x^{-1}}{-1}+c$

$=\frac{x^2}{2}+\frac{1}{x}+c$

(ii) ${\int }^{}\left(x^{\frac{2}{3}}+1\right)dx$

$={\int }^{}{x}^{\frac{2}{3}}dx+{\int }^{}dx$

$=\frac{3}{5}{x}^{\frac{5}{3}}+x+c$

(iii) ${\int }^{}\left(x^{\frac{3}{2}}+2e^x-\frac{1}{x}\right)dx$

$={\int }^{}{x}^{\frac{3}{2}}dx+{\int }^{}2e^{x}dx-{\int }^{}\frac{1}{x}dx$

$=\frac{2}{5}{x}^{\frac{5}{2}}+2e^x-\log \left|x\right|+c$