40) $f(x) = \dfrac{2x^2 - 3}{x}$
$F(x)=\displaystyle\int\left(\dfrac{2x^2 - 3}{x}\right)dx$
$\to F(x) = 2\displaystyle\int xdx - 3\displaystyle\int\dfrac{dx}{x}$
$\to F(x) = x^2 - 3\ln|x| + C$
Ta có:
$F(1) = \dfrac72$
$\to 1^2 -3\ln1 + C = \dfrac72$
$\to 1 + C = \dfrac72$
$\to C = \dfrac52$
Vậy $F(x) = x^2 - 3\ln|x| +\dfrac52$
41) $f(x) = \sin4x.\cos3x$
$F(x) = \displaystyle\int\sin4x.\cos3xdx$
$\to F(x) =\dfrac12\displaystyle\int(\sin7x + \sin x)dx$
$\to F(x) =\dfrac12\displaystyle\int\sin7xdx + \dfrac12\displaystyle\int\sin xdx$
$\to F(x) =\dfrac12\cdot \dfrac{-\cos7x}{7} +\dfrac12\cdot(-\cos x) + C$
$\to F(x) = -\dfrac{1}{14}\cos7x -\dfrac12\cos x + C$
Ta có:
$F\left(\dfrac{\pi}{2}\right) = 1$
$\to -\dfrac{1}{14}\cos\dfrac{7\pi}{2} -\dfrac12\cos\dfrac{\pi}{2} + C = 1$
$\to -\dfrac{1}{14}\cdot0-\dfrac12\cdot0 + C = 1$
$\to C = 1$
Vậy $F(x) = -\dfrac{1}{14}\cos7x -\dfrac12\cos x + 1$
