integral from a to 2a of ax-(a^3)/x

解答

\int_{a}^{2 a} a x - \frac{a ^{3}}{x} d x

a (2 a^{2} - \frac{a ^{2}}{2}) - ln (2) a^{3}

求解步骤

\int_{a}^{2 a} a x - \frac{a ^{3}}{x} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{a}^{2 a} a x d x - \int_{a}^{2 a} \frac{a ^{3}}{x} d x

\int_{a}^{2 a} a x d x = a (2 a^{2} - \frac{a ^{2}}{2})

\int_{a}^{2 a} \frac{a ^{3}}{x} d x = ln (2) a^{3}

= a (2 a^{2} - \frac{a ^{2}}{2}) - ln (2) a^{3}

\int \frac{1}{( x ^{2} - 2 x + 5 ) ^{3}} d x

\int \frac{1}{x ( x - 1 )} d x

x y^{^{'}} + 2 y = 2 e x

\frac{d}{d x} (sec^{2} (3 x))

y^{^{''}} - 2 \cdot y^{^{'}} = sin (x)