integral of 1/(sqrt(12+x^2))

解答

\int \frac{1}{12 + x ^{2}} d x

ln (\frac{1}{2 3} x + 12 + x^{2}) + C

求解步骤

\int \frac{1}{12 + x ^{2}} d x

使用三角换元法

= \int sec (u) d u

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= ln ∣ tan (u) + sec (u) ∣

u = arctan (\frac{1}{2 3} x)

代回

= ln tan (arctan (\frac{1}{2 3} x)) + sec (arctan (\frac{1}{2 3} x))

化简

ln tan (arctan (\frac{1}{2 3} x)) + sec (arctan (\frac{1}{2 3} x)) : ln (\frac{1}{2 3} x + 12 + x^{2})

= ln (\frac{1}{2 3} x + 12 + x^{2})

解答补常数

= ln (\frac{1}{2 3} x + 12 + x^{2}) + C