integral of 5xarctan(x)

解答

\int 5 x arctan (x) d x

5 (\frac{1}{2} x^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x)) + C

求解步骤

\int 5 x arctan (x) d x

提出常数:

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

= 5 \cdot \int x arctan (x) d x

使用分布积分法

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

\int \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x = \frac{1}{2} (- arctan (x) + x)

= 5 (\frac{1}{2} x^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x))

解答补常数

= 5 (\frac{1}{2} x^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x)) + C

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