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受欢迎的微积分 >

integral of (10x)/(4x^2+8x+9)

解答

\int \frac{10 x}{4 x ^{2} + 8 x + 9} d x

解答

\frac{5}{4} ln \frac{4}{5} (x + 1)^{2} + 1 - 5 arctan (\frac{2}{5} (x + 1)) + C

求解步骤

\int \frac{10 x}{4 x ^{2} + 8 x + 9} d x

提出常数:

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

= 10 \cdot \int \frac{x}{4 x ^{2} + 8 x + 9} d x

对

4 x^{2} + 8 x + 9

配方:

4 (x + 1)^{2} + 5

= 10 \cdot \int \frac{x}{4 ( x + 1 ) ^{2} + 5} d x

使用换元积分法

= 10 \cdot \int \frac{u - 1}{4 u ^{2} + 5} d u

使用换元积分法

= 10 \cdot \int \frac{5 v - 2}{4 5 ( v ^{2} + 1 )} d v

提出常数:

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

= 10 \cdot \frac{1}{4 5} \cdot \int \frac{5 v - 2}{v ^{2} + 1} d v

乘开

\frac{5 v - 2}{v ^{2} + 1} : \frac{5 v}{v ^{2} + 1} - \frac{2}{v ^{2} + 1}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 10 \cdot \frac{1}{4 5} (\int \frac{5 v}{v ^{2} + 1} d v - \int \frac{2}{v ^{2} + 1} d v)

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

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

= 10 \cdot \frac{1}{4 5} (\frac{5}{2} ln v^{2} + 1 - 2 arctan (v))

代回

= 10 \cdot \frac{1}{4 5} (\frac{5}{2} ln (\frac{2}{5} (x + 1))^{2} + 1 - 2 arctan (\frac{2}{5} (x + 1)))

化简

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

= \frac{5}{4} ln \frac{4}{5} (x + 1)^{2} + 1 - 5 arctan (\frac{2}{5} (x + 1))

解答补常数

= \frac{5}{4} ln \frac{4}{5} (x + 1)^{2} + 1 - 5 arctan (\frac{2}{5} (x + 1)) + C

作图

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