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受欢迎的微积分 >

integral of (arctan(7x))/(1+49x^2)

解答

\int \frac{arctan ( 7 x )}{1 + 49 x ^{2}} d x

解答

\frac{1}{14} arctan^{2} (7 x) + C

求解步骤

\int \frac{arctan ( 7 x )}{1 + 49 x ^{2}} d x

使用换元积分法

= \int \frac{arctan ( u )}{7 ( 1 + u ^{2} )} d u

提出常数:

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

= \frac{1}{7} \cdot \int \frac{arctan ( u )}{1 + u ^{2}} d u

使用换元积分法

= \frac{1}{7} \cdot \int v d v

使用幂法则

= \frac{1}{7} \cdot \frac{v ^{2}}{2}

代回

= \frac{1}{7} \cdot \frac{arctan ^{2} ( 7 x )}{2}

化简

\frac{1}{7} \cdot \frac{arctan ^{2} ( 7 x )}{2} : \frac{1}{14} arctan^{2} (7 x)

= \frac{1}{14} arctan^{2} (7 x)

解答补常数

= \frac{1}{14} arctan^{2} (7 x) + C

作图

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