integral of (20)/((x-4)(x^2+4))

解答

\int \frac{20}{( x - 4 ) ( x ^{2} + 4 )} d x

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

求解步骤

\int \frac{20}{( x - 4 ) ( x ^{2} + 4 )} d x

提出常数:

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

= 20 \cdot \int \frac{1}{( x - 4 ) ( x ^{2} + 4 )} d x

将

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

用部份分式展开:

\frac{1}{20 ( x - 4 )} + \frac{- x - 4}{20 ( x ^{2} + 4 )}

= 20 \cdot \int \frac{1}{20 ( x - 4 )} + \frac{- x - 4}{20 ( x ^{2} + 4 )} d x

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

= 20 (\int \frac{1}{20 ( x - 4 )} d x + \int \frac{- x - 4}{20 ( x ^{2} + 4 )} d x)

\int \frac{1}{20 ( x - 4 )} d x = \frac{1}{20} ln ∣ x - 4 ∣

\int \frac{- x - 4}{20 ( x ^{2} + 4 )} d x = \frac{1}{20} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}))

= 20 (\frac{1}{20} ln ∣ x - 4 ∣ + \frac{1}{20} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})))

化简

20 (\frac{1}{20} ln ∣ x - 4 ∣ + \frac{1}{20} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}))) : ln ∣ x - 4 ∣ - \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})

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

解答补常数

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

t a y l or sin (x), \frac{3 π}{2}

\frac{d}{d x} (100 x - x^{2})

n = 1 \sum \infty 3 n e^{- n^{2}}

\int (1 + 9 - x^{2}) d x

\frac{d}{d x} (4 x^{3} - 6 x)