Giải bài 9 trang 9

Đề bài

Tìm nguyên hàm của các hàm số sau:

a) $f\left(x\right)=2x^2-4x^5+6$;

b) $f\left(x\right)=\left(x+3\right)\left(-2-x\right)$;

c) $f\left(x\right)=\frac{x^6-7x^3}{x}\left(x>0\right)$.

Lời giải chi tiết

a) $\int f\left(x\right)dx=\int \left(2x^2-4x^5+6\right)dx$

$=\int 2x^{2}dx-\int 4x^{5}dx+\int 6dx$

$=\frac{2}{3}\int 3x^{2}dx-\frac{2}{3}\int 6x^{5}dx+6\int 1dx$

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

$=\frac{2}{3}{x}^3-\frac{2}{3}{x}^6+6x+C$.

b) $f\left(x\right)=\left(x+3\right)\left(-2-x\right)$

$=-2x-6-x^2-3x=-x^2-5x-6$.

$\int f\left(x\right)dx=\int \left(-x^2-5x-6\right)dx$

$=-\int x^{2}dx-\int 5xdx-\int 6dx$

$=-\frac{1}{3}\int 3x^{2}dx-\frac{5}{2}\int 2xdx-6\int 1dx$

$=-\frac{1}{3}\int {\left(x^3\right)}^{\prime }dx-\frac{5}{2}\int {\left(x^2\right)}^{\prime }dx-6\int {\left(x\right)}^{\prime }dx$

$=-\frac{1}{3}{x}^3-\frac{5}{2}{x}^2-6x+C$.

c) $f\left(x\right)=\frac{x^6-7x^3}{x}=\frac{x^6}{x}-\frac{7x^3}{x}=x^5-7x^2$.

$\int f\left(x\right)dx=\int \left(x^5-7x^2\right)dx$

$=\int x^{5}dx-\int 7x^{2}dx$

$=\frac{1}{6}\int 6x^{5}dx-\frac{7}{3}\int 3x^{2}dx$

$=\frac{1}{6}\int {\left(x^6\right)}^{\prime }dx-\frac{7}{3}\int {\left(x^3\right)}^{\prime }dx$

$=\frac{1}{6}{x}^6-\frac{7}{3}{x}^3+C$.