Giải bài tập 16 trang 29

Đề bài

Tính các tích phân sau:

a) $\underset{0}{\overset{1}{\int }}\left(4x^3+x\right)dx$;

b) $\underset{1}{\overset{2}{\int }}\frac{x-2}{x^2}dx$;

c) $\underset{0}{\overset{4}{\int }}{2}^{2x}dx$;

d) $\underset{1}{\overset{2}{\int }}\left(e^{x-1}+2^{x+1}\right)dx$.

Lời giải chi tiết

a) $\underset{0}{\overset{1}{\int }}\left(4x^3+x\right)dx={\left(x^4+\frac{x^2}{2}\right)|}_0^1=\frac{3}{2}-0=\frac{3}{2}$.

b) $\underset{1}{\overset{2}{\int }}\frac{x-2}{x^2}dx=\underset{1}{\overset{2}{\int }}\left(\frac{1}{x}-\frac{2}{x^2}\right)dx$

$=\underset{1}{\overset{2}{\int }}\left(\frac{1}{x}-2x^{-2}\right)dx={\left(\ln \left|x\right|-2\frac{x^{-1}}{-1}\right)|}_1^2$

$={\left(\ln \left|x\right|+\frac{2}{x}\right)|}_1^2=\left(\ln 2+1\right)-\left(\ln 1+2\right)=\ln 2-1$.

c) $\underset{0}{\overset{4}{\int }}{2}^{2x}dx=\underset{0}{\overset{4}{\int }}{4}^{x}dx={\left(\frac{4^x}{\ln 4}\right)|}_0^4=\frac{4^4}{\ln 4}-\frac{4^0}{\ln 4}=\frac{255}{\ln 4}$.

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

$=\frac{1}{e}\underset{1}{\overset{2}{\int }}{e}^{x}dx+2\underset{1}{\overset{2}{\int }}{2}^{x}dx=\frac{1}{e}.{\left(e^x\right)|}_1^2+2.{\left(\frac{2^x}{\ln 2}\right)|}_1^2$

$=\frac{1}{e}\left(e^2-e^1\right)+2.\left(\frac{2^2}{\ln 2}-\frac{2^1}{\ln 2}\right)$

$=e-1+2.\frac{2}{\ln 2}=e-1+\frac{4}{\ln 2}$.