integral of ((4x+3))/(4x^3+8x^2+3x)

解答

\int \frac{( 4 x + 3 )}{4 x ^{3} + 8 x ^{2} + 3 x} d x

ln ∣ x ∣ - \frac{1}{2} ln ∣ 2 x + 1 ∣ - \frac{1}{2} ln ∣ 2 x + 3 ∣ + C

求解步骤

\int \frac{4 x + 3}{4 x ^{3} + 8 x ^{2} + 3 x} d x

将

\frac{4 x + 3}{4 x ^{3} + 8 x ^{2} + 3 x}

用部份分式展开:

\frac{1}{x} - \frac{1}{2 x + 1} - \frac{1}{2 x + 3}

= \int \frac{1}{x} - \frac{1}{2 x + 1} - \frac{1}{2 x + 3} 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

= \int \frac{1}{x} d x - \int \frac{1}{2 x + 1} d x - \int \frac{1}{2 x + 3} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{1}{2 x + 1} d x = \frac{1}{2} ln ∣ 2 x + 1 ∣

\int \frac{1}{2 x + 3} d x = \frac{1}{2} ln ∣ 2 x + 3 ∣

= ln ∣ x ∣ - \frac{1}{2} ln ∣ 2 x + 1 ∣ - \frac{1}{2} ln ∣ 2 x + 3 ∣

解答补常数

= ln ∣ x ∣ - \frac{1}{2} ln ∣ 2 x + 1 ∣ - \frac{1}{2} ln ∣ 2 x + 3 ∣ + C

\int sec^{4} (7 x) d x

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

\int 24 x (2 x^{2} + 5)^{2} d x

\int (8 x^{3} + 3 3 x^{2}) d x

\int \frac{e ^{x}}{( e ^{x} + 1 ) ^{14}} d x