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

解答

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

\frac{1}{2} arctan (2 x) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= \frac{1}{2} arctan (u)

u = 2 x

代回

= \frac{1}{2} arctan (2 x)

解答补常数

= \frac{1}{2} arctan (2 x) + C