integral of (x^2+2x)*3^x

解答

\int (x^{2} + 2 x) \cdot 3^{x} d x

\frac{3 ^{x} x ^{2}}{ln ( 3 )} + \frac{- 2 + 2 ln ( 3 )}{ln ( 3 )} (\frac{3 ^{x} x}{ln ( 3 )} - \frac{3 ^{x}}{ln ^{2} ( 3 )}) + C

求解步骤

\int (x^{2} + 2 x) \cdot 3^{x} d x

乘开

(x^{2} + 2 x) \cdot 3^{x} : 3^{x} x^{2} + 2 \cdot 3^{x} x

= \int 3^{x} x^{2} + 2 \cdot 3^{x} 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 3^{x} x^{2} d x + \int 2 \cdot 3^{x} x d x

\int 3^{x} x^{2} d x = \frac{3 ^{x} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{x} x}{ln ( 3 )} - \frac{3 ^{x}}{ln ^{2} ( 3 )})

\int 2 \cdot 3^{x} x d x = 2 (\frac{3 ^{x} x}{ln ( 3 )} - \frac{3 ^{x}}{ln ^{2} ( 3 )})

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

\int_{- 3}^{3} (13 - x^{2}) - (x^{2} - 5) d x

\int \frac{1}{x ^{2} + 169} d x

(2 x + 1) (\frac{d y}{d x}) + y = (2 x + 1)^{\frac{3}{2}}

y^{^{''}} + 3 y = - 48 x^{2} e^{3 x}

f (x) = arcsinh (x)