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受欢迎的微积分 >

integral of (7x)/((x^2+8)^2+25)

解答

\int \frac{7 x}{( x ^{2} + 8 ) ^{2} + 25} d x

解答

\frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8)) + C

求解步骤

\int \frac{7 x}{( x ^{2} + 8 ) ^{2} + 25} d x

提出常数:

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

= 7 \cdot \int \frac{x}{( x ^{2} + 8 ) ^{2} + 25} d x

乘开

(x^{2} + 8)^{2} + 25 : x^{4} + 16 x^{2} + 89

= 7 \cdot \int \frac{x}{x ^{4} + 16 x ^{2} + 89} d x

使用换元积分法

= 7 \cdot \int \frac{1}{4 u ( 16 u + u + 89 )} d u

提出常数:

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

= 7 \cdot \frac{1}{4} \cdot \int \frac{1}{u ( 16 u + u + 89 )} d u

使用换元积分法

= 7 \cdot \frac{1}{4} \cdot \int \frac{2}{16 v + v ^{2} + 89} d v

提出常数:

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

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{16 v + v ^{2} + 89} d v

对

16 v + v^{2} + 89

配方:

(v + 8)^{2} + 25

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{( v + 8 ) ^{2} + 25} d v

使用换元积分法

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{w ^{2} + 25} d w

使用换元积分法

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{5 ( t ^{2} + 1 )} d t

提出常数:

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

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} \cdot \int \frac{1}{t ^{2} + 1} d t

使用积分常见定则:

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

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} arctan (t)

代回

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} arctan (\frac{x ^{4} + 8}{5})

化简

7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} arctan (\frac{x ^{4} + 8}{5}) : \frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8))

= \frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8))

解答补常数

= \frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8)) + C

作图

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