37) $f(x) = x^4 - 5x^2 + 2$
$F(x) = \displaystyle\int f(x)dx$
$\to F(x) = \displaystyle\int(x^4 - 5x^2 + 2)dx$
$\to F(x) = \displaystyle\int x^4dx - 5\displaystyle\int x^2xdx + 2\displaystyle\int dx$
$\to F(x) = \dfrac15x^5 - \dfrac53x^3 + 2x + C$
Ta có:
$F(0) = -2$
$\to \dfrac15\cdot0^5 - \dfrac53\cdot0^3 + 2.0+ C = -2$
$\to C = -2$
Vậy $F(x) = \dfrac15x^5 - \dfrac53x^3 + 2x -2$
38) $f(x) = 3-5\sin x$
$F(x) = \displaystyle\int f(x)dx$
$\to F(x) = \displaystyle\int(3 - 5\sin x)dx$
$\to F(x) = 3\displaystyle\int dx - 5\displaystyle\int\sin xdx$
$\to F(x) = 3x +5\cos x + C$
Ta có:
$F\left(\dfrac{\pi}{4}\right) = 1$
$\to 3\cdot\dfrac{\pi}{4} + 5\cos\dfrac{\pi}{4} + C = 1$
$\to \dfrac{3\pi}{4} + \dfrac{5\sqrt2}{2} + C = 1$
$\to C = 1 - \dfrac{3\pi}{4} - \dfrac{5\sqrt2}{2}$
Vậy $F(x) = 3x +5\cos x + 1 - \dfrac{3\pi}{4} - \dfrac{5\sqrt2}{2}$
39) $f(x) = \dfrac{1-2x^2}{x}$
$F(x) = \displaystyle\int f(x)dx$
$\to F(x) = \displaystyle\int\left(\dfrac{1 -2x^2}{x}\right)dx$
$\to F(x) = \displaystyle\int\dfrac{dx}{x} - 2\displaystyle\int xdx$
$\to F(x) = \ln|x| - x^2 + C$
Ta có:
$F\left(e^2\right) = 2$
$\to \ln\left|e^2\right| - e^4 + C = 2$
$\to 2 - e^4 + C = 2$
$\to C = e^4$
Vậy $F(x) = \ln|x| - x^2 + e^4$
