integral of (2)5^{x^2+7}

解

\int (2) 5^{x^{2} + 7} d x

\frac{78125 π}{ln ( 5 )} erfi (ln (5) x) + C

解答ステップ

\int 2 \cdot 5^{x^{2} + 7} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int 5^{x^{2} + 7} d x

5^{x^{2} + 7} = 5^{7} \cdot 5^{x^{2}}

= 2 \cdot \int 5^{7} \cdot 5^{x^{2}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot 5^{7} \cdot \int 5^{x^{2}} d x

簡素化

= 2 \cdot 78125 \cdot \int 5^{x^{2}} d x

非初等積分を使用する:

\int 5^{x^{2}} d x = \frac{π erfi ( x ln ( 5 ) )}{2 ln ( 5 )}

= 2 \cdot 78125 \cdot \frac{π erfi ( x ln ( 5 ) )}{2 ln ( 5 )}

簡素化

2 \cdot 78125 \cdot \frac{π erfi ( x ln ( 5 ) )}{2 ln ( 5 )} : \frac{78125 π}{ln ( 5 )} erfi (ln (5) x)

= \frac{78125 π}{ln ( 5 )} erfi (ln (5) x)

定数を解答に追加する

= \frac{78125 π}{ln ( 5 )} erfi (ln (5) x) + C