Giải bài tập 4 trang 40

Đề bài

Tìm:

a) $\int 2x(x^3-x+2)dx$

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

c) $\int \left(3+2{\tan }^{2}x\right)dx$

d) $\int \left(1-3{\cot }^{2}x\right)dx$

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

g) $\int \left({2.6}^{2x}-e^{-x+1}\right)dx$

Lời giải chi tiết

a) $\int 2x(x^3-x+2)dx=\int \left(2x^4-2x^2+4x\right)dx$

$=\frac{2x^5}{5}-\frac{2x^3}{3}+2x^2+C$.

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

c) $\int \left(3+2{\tan }^{2}x\right)dx=\int \left(1+2(1+{\tan }^{2}x)\right)dx$

$=\int (1+\frac{2}{{\cos }^{2}x})dx=x+2\tan x+C$.

d) $\int \left(1-3{\cot }^{2}x\right)dx=\int (4-3(1+{\cot }^{2}x\left)\right)dx$

$=\int \left(4-\frac{3}{{\sin }^{2}x}\right)dx=4x+3\cot x+C$.

e) $\int \left(\sin +2^{-x+1}\right)dx=-\cos x-\frac{2^{-x+1}}{\ln 2}+C$.

g) $\int \left({2.6}^{2x}-e^{-x+1}\right)dx=\frac{6^{2x}}{\ln 6}-e^{-x+1}+C$.