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受欢迎的微积分 >

integral of 5/(36+(x-1)^2)

解答

\int \frac{5}{36 + ( x - 1 ) ^{2}} d x

解答

\frac{5}{6} arctan (\frac{1}{6} (x - 1)) + C

求解步骤

\int \frac{5}{36 + ( x - 1 ) ^{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{1}{36 + ( x - 1 ) ^{2}} d x

使用换元积分法

= 5 \cdot \int \frac{1}{36 + u ^{2}} d u

使用换元积分法

= 5 \cdot \int \frac{1}{6 ( v ^{2} + 1 )} d v

提出常数:

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

= 5 \cdot \frac{1}{6} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

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

= 5 \cdot \frac{1}{6} arctan (v)

代回

= 5 \cdot \frac{1}{6} arctan (\frac{x - 1}{6})

化简

5 \cdot \frac{1}{6} arctan (\frac{x - 1}{6}) : \frac{5}{6} arctan (\frac{1}{6} (x - 1))

= \frac{5}{6} arctan (\frac{1}{6} (x - 1))

解答补常数

= \frac{5}{6} arctan (\frac{1}{6} (x - 1)) + C

作图

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