Giải thích các bước giải:
Câu 1:
Ta có:
$f(x)=\displaystyle\int x\sqrt{1-x^2}dx$
$\to f(x)=\dfrac12\displaystyle\int 2x\sqrt{1-x^2}dx$
$\to f(x)=\dfrac12\displaystyle\int \sqrt{1-x^2}d(x^2)$
$\to f(x)=-\dfrac12\displaystyle\int \sqrt{1-x^2}d(1-x^2)$
$\to f(x)=-\dfrac12\displaystyle\int (1-x^2)^{\frac12}d(1-x^2)$
$\to f(x)=-\dfrac12\cdot \dfrac1{\dfrac12+1}\cdot(1-x^2)^{\frac12+1}+C$
$\to f(x)=-\dfrac12\cdot \dfrac23\cdot(1-x^2)^{\frac32}+C$
$\to f(x)=-\dfrac13\cdot(1-x^2)^{\frac32}+C$
Câu 3:
Ta có:
$3x^2f(x)+x^3f'(x)=2$
$\to \displaystyle\int 3x^2f(x)+x^3f'(x)dx=\displaystyle\int 2dx$
$\to x^3f(x)=2x+C$
Ta có $f(-1)=0$
$\to (-1)^3\cdot f(-1)=2\cdot (-1)+C$
$\to (-1)^3\cdot 0=2\cdot (-1)+C$
$\to C=2$
$\to x^3f(x)=2x+2$
$\to 2^3\cdot f(2)=2\cdot 2+2$
$\to f(2)=\dfrac34$
