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受欢迎的微积分 >

integral of (-5arctan(x))/(2+2x^2)

解答

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

解答

- \frac{5}{4} arctan^{2} (x) + C

求解步骤

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

提出常数:

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

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

使用换元积分法

= - 5 \cdot \int \frac{u}{2} d u

提出常数:

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

= - 5 \cdot \frac{1}{2} \cdot \int u d u

使用幂法则

= - 5 \cdot \frac{1}{2} \cdot \frac{u ^{2}}{2}

u = arctan (x)

代回

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

化简

- 5 \cdot \frac{1}{2} \cdot \frac{arctan ^{2} ( x )}{2} : - \frac{5}{4} arctan^{2} (x)

= - \frac{5}{4} arctan^{2} (x)

解答补常数

= - \frac{5}{4} arctan^{2} (x) + C

作图

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