Giải bài 23 trang 15

Đề bài

Tìm:

a) $\int x^{\frac{1}{3}}dx$;

b) $\int \sqrt{\frac{1}{x^7}}dx$;

c) $\int \frac{1}{\sqrt[3]{3x^{\frac{4}{5}}}}dx$;

d) $\int {\left(x-\frac{1}{x}\right)}^{2}dx$;

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

g) $\int \left(3x^2-\frac{4}{x}\right)\left(2x+5\right)dx$.

Lời giải chi tiết

a) $\int x^{\frac{1}{3}}dx=\frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}+C$

$=\frac{x^{\frac{4}{3}}}{\frac{4}{3}}+C=\frac{3}{4}{x}^{\frac{4}{3}}+C$.

b) $\int \sqrt{\frac{1}{x^7}}dx=\int \frac{1}{\sqrt{x^7}}dx=\int \frac{1}{x^{\frac{7}{2}}}dx=\int x^{-\frac{7}{2}}dx$

$=\frac{x^{-\frac{7}{2}+1}}{-\frac{7}{2}+1}+C=\frac{x^{-\frac{5}{2}}}{-\frac{5}{2}}+C=-\frac{2}{5}{x}^{-\frac{5}{2}}+C$.

c) $\int \frac{1}{\sqrt[3]{3x^{\frac{4}{5}}}}dx=\int \frac{1}{x^{\frac{4}{15}}}dx=\int x^{-\frac{4}{15}}dx$

$=\frac{x^{-\frac{4}{15}+1}}{-\frac{4}{15}+1}+C=\frac{x^{\frac{11}{15}}}{\frac{11}{15}}+C=\frac{15}{11}{x}^{\frac{11}{15}}+C$.

d) $\int {\left(x-\frac{1}{x}\right)}^{2}dx=\int \left(x^2-2.x.\frac{1}{x}+\frac{1}{x^2}\right)dx$

$=\int \left(x^2-2+x^{-2}\right)dx=\frac{x^{2+1}}{2+1}-2x+\frac{x^{-2+1}}{-2+1}+C$

$=\frac{x^3}{3}-2x-\frac{1}{x}+C$.

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

$=\int \left(x-2-\frac{3}{x}\right)dx=\frac{x^2}{2}-2x-3\ln \left|x\right|+C$.

g) $\int \left(3x^2-\frac{4}{x}\right)\left(2x+5\right)dx$

$=\int \left(6x^3+15x^2-8-\frac{20}{x}\right)dx$

$=\frac{6x^4}{4}+\frac{15x^3}{3}-8x-20\ln \left|x\right|+C$

$=\frac{3}{2}{x}^4+5x^3-8x-20\ln \left|x\right|+C$.