integral of 1/(sqrt(-x^2-14x-48))

解答

\int \frac{1}{- x ^{2} - 14 x - 48} d x

arcsin (x + 7) + C

求解步骤

\int \frac{1}{- x ^{2} - 14 x - 48} d x

对

- x^{2} - 14 x - 48

配方:

- (x + 7)^{2} + 1

= \int \frac{1}{- ( x + 7 ) ^{2} + 1} d x

使用换元积分法

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

使用积分常见定则:

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

= arcsin (u)

u = x + 7

代回

= arcsin (x + 7)

解答补常数

= arcsin (x + 7) + C

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