Giải bài 4.13 trang 12

Đề bài

Tính các tích phân sau:

a) $\underset{0}{\overset{1}{\int }}{\left(1-2x\right)}^{2}dx$;

b) $\underset{1}{\overset{4}{\int }}\frac{x-2}{\sqrt{x}}dx$.

Lời giải chi tiết

a) Ta có $\underset{0}{\overset{1}{\int }}{\left(1-2x\right)}^{2}dx=\underset{0}{\overset{1}{\int }}\left(1-4x+4x^2\right)dx=\underset{0}{\overset{1}{\int }}dx-4\underset{0}{\overset{1}{\int }}xdx+4\underset{0}{\overset{1}{\int }}{x}^{2}dx$

$={x|}_0^1-4{\cdot \frac{x^2}{2}x|}_0^1+4\cdot {\frac{x^3}{3}|}_0^1=1-2\left(1-0\right)+\frac{4}{3}\left(1-0\right)=1-2+\frac{4}{3}=\frac{1}{3}$.

b) Ta có $\underset{1}{\overset{4}{\int }}\frac{x-2}{\sqrt{x}}dx=\underset{1}{\overset{4}{\int }}\left(\sqrt{x}-\frac{2}{\sqrt{x}}\right)dx=\underset{1}{\overset{4}{\int }}\sqrt{x}dx-2\underset{1}{\overset{4}{\int }}\frac{1}{\sqrt{x}}dx$ $={\frac{x\sqrt{x}}{\left(\frac{3}{2}\right)}|}_1^4-2\cdot 2{\sqrt{x}|}_1^4$

$=\frac{2}{3}\left(4\sqrt{4}-1\right)-4\left(\sqrt{4}-1\right)=\frac{14}{3}-4=\frac{2}{3}$.