integral of 2/(1/2+x^2)

解答

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

22 arctan (2 x) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

= 22 \cdot \int \frac{1}{1 + u ^{2}} d u

使用积分常见定则:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 22 arctan (u)

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

代回

= 22 arctan \frac{x}{\frac{1}{2}}

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

= 22 arctan (2 x)

解答补常数

= 22 arctan (2 x) + C

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

\int \frac{4 x ^{2}}{x ^{3} + 16} d x

d er i v a t i v e y = x^{\frac{5}{3}} - x^{\frac{2}{3}}

x \to - 4 lim (x (x + 3) (x + 2))

t an g e n t \frac{x}{- 6 x + 6}