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受欢迎的微积分 >

integral of (3x)/(x^2-x-6)

解答

\int \frac{3 x}{x ^{2} - x - 6} d x

解答

6 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + C

求解步骤

\int \frac{3 x}{x ^{2} - x - 6} d x

提出常数:

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

= 3 \cdot \int \frac{x}{x ^{2} - x - 6} d x

对

x^{2} - x - 6

配方:

(x - \frac{1}{2})^{2} - \frac{25}{4}

= 3 \cdot \int \frac{x}{( x - \frac{1}{2} ) ^{2} - \frac{25}{4}} d x

使用换元积分法

= 3 \cdot \int \frac{2 ( 2 u + 1 )}{4 u ^{2} - 25} d u

提出常数:

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

= 3 \cdot 2 \cdot \int \frac{2 u + 1}{4 u ^{2} - 25} d u

乘开

\frac{2 u + 1}{4 u ^{2} - 25} : \frac{2 u}{4 u ^{2} - 25} + \frac{1}{4 u ^{2} - 25}

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

= 3 \cdot 2 (\int \frac{2 u}{4 u ^{2} - 25} d u + \int \frac{1}{4 u ^{2} - 25} d u)

\int \frac{2 u}{4 u ^{2} - 25} d u = \frac{1}{4} ln 4 u^{2} - 25

\int \frac{1}{4 u ^{2} - 25} d u = - \frac{1}{20} (ln \frac{2}{5} u + 1 - ln \frac{2}{5} u - 1)

= 3 \cdot 2 (\frac{1}{4} ln 4 u^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} u + 1 - ln \frac{2}{5} u - 1))

u = x - \frac{1}{2}

代回

= 3 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{1}{2})^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} (x - \frac{1}{2}) + 1 - ln \frac{2}{5} (x - \frac{1}{2}) - 1))

化简

3 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{1}{2})^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} (x - \frac{1}{2}) + 1 - ln \frac{2}{5} (x - \frac{1}{2}) - 1)) : 6 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x))

= 6 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x))

解答补常数

= 6 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + C

作图

流行的例子

x^2(dy)/(dx)+3xy=ln(x)

x^{2} \frac{d y}{d x} + 3 x y = ln (x)

implicit xy=cot(xy)

im pl i c i t x y = cot (x y)

integral of x/((z^2+x^2)^{3/2)}

\int \frac{x}{( z ^{2} + x ^{2} ) ^{\frac{3}{2}}} d x

derivative y= 3/x

d er i v a t i v e y = \frac{3}{x}

integral of (6)

\int (6) d x