integral from 1 to 2 of 5/(x^3+2x)

解答

\int_{1}^{2} \frac{5}{x ^{3} + 2 x} d x

\frac{5}{4} ln (2)

十进制

0.86643 \dots

求解步骤

\int_{1}^{2} \frac{5}{x ^{3} + 2 x} d x

提出常数:

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

= 5 \cdot \int_{1}^{2} \frac{1}{x ^{3} + 2 x} d x

将

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

用部份分式展开:

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

= 5 \cdot \int_{1}^{2} \frac{1}{2 x} - \frac{x}{2 ( x ^{2} + 2 )} 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

= 5 (\int_{1}^{2} \frac{1}{2 x} d x - \int_{1}^{2} \frac{x}{2 ( x ^{2} + 2 )} d x)

\int_{1}^{2} \frac{1}{2 x} d x = \frac{1}{2} ln (2)

\int_{1}^{2} \frac{x}{2 ( x ^{2} + 2 )} d x = \frac{1}{4} ln (2)

= 5 (\frac{1}{2} ln (2) - \frac{1}{4} ln (2))

化简

5 (\frac{1}{2} ln (2) - \frac{1}{4} ln (2)) : \frac{5}{4} ln (2)

= \frac{5}{4} ln (2)

\int_{0}^{1} (5^{x} - 3^{x}) d x

t an g e n t y = 2 x^{3}, (1, 2)

d er i v a t i v e f (x) = 6 x + 3

\int \frac{ln ( 11 x )}{x ^{9}} d x

\int \frac{( 7 x ^{2} + 5 x - 6 )}{( x ^{2} - 4 ) ( x + 2 )} d x