integral of (0.1q+1)e^{0.3q}

解答

\int (0.1 q + 1) e^{0.3 q} d q

0.33333 \dots e^{0.3 q} q + 2.22222 \dots e^{0.3 q} + C

求解步骤

\int (0.1 q + 1) e^{0.3 q} d q

乘开

(0.1 q + 1) e^{0.3 q} : 0.1 e^{0.3 q} q + e^{0.3 q}

= \int 0.1 e^{0.3 q} q + e^{0.3 q} d q

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 0.1 e^{0.3 q} q d q + \int e^{0.3 q} d q

\int 0.1 e^{0.3 q} q d q = 0.1 (3.33333 \dots e^{0.3 q} q - 11.11111 \dots e^{0.3 q})

\int e^{0.3 q} d q = 3.33333 \dots e^{0.3 q}

= 0.1 (3.33333 \dots e^{0.3 q} q - 11.11111 \dots e^{0.3 q}) + 3.33333 \dots e^{0.3 q}

化简

0.1 (3.33333 \dots e^{0.3 q} q - 11.11111 \dots e^{0.3 q}) + 3.33333 \dots e^{0.3 q} : 0.33333 \dots e^{0.3 q} q + 2.22222 \dots e^{0.3 q}

= 0.33333 \dots e^{0.3 q} q + 2.22222 \dots e^{0.3 q}

解答补常数

= 0.33333 \dots e^{0.3 q} q + 2.22222 \dots e^{0.3 q} + C

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