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受欢迎的微积分 >

integral of (2x-2)/((x-1)^3+4(x-1))

解答

\int \frac{2 x - 2}{( x - 1 ) ^{3} + 4 ( x - 1 )} d x

解答

arctan (\frac{1}{2} (x - 1)) + C

求解步骤

\int \frac{2 x - 2}{( x - 1 ) ^{3} + 4 ( x - 1 )} d x

将

\frac{2 x - 2}{( x - 1 ) ^{3} + 4 ( x - 1 )}

用部份分式展开:

\frac{2}{x ^{2} - 2 x + 5}

= \int \frac{2}{x ^{2} - 2 x + 5} d x

提出常数:

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

= 2 \cdot \int \frac{1}{x ^{2} - 2 x + 5} d x

对

x^{2} - 2 x + 5

配方:

(x - 1)^{2} + 4

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

使用换元积分法

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

使用换元积分法

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

提出常数:

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

= 2 \cdot \frac{1}{2} \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)

= 2 \cdot \frac{1}{2} arctan (v)

代回

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

化简

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

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

解答补常数

= arctan (\frac{1}{2} (x - 1)) + C

作图

流行的例子

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\int \frac{3}{x ^{2} + 2 x + 1} d x

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

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\int y sin^{2} (x) d x

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\int x^{4} arcsin (x) d x

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\int (x^{3} + 5 x - 1) e^{x} d x