integral of (x+3)/(sqrt(4-(x-2)^2))

解答

\int \frac{x + 3}{4 - ( x - 2 ) ^{2}} d x

- - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1) + C

求解步骤

\int \frac{x + 3}{4 - ( x - 2 ) ^{2}} d x

使用换元积分法

= \int \frac{u + 5}{4 - u ^{2}} d u

使用三角换元法

= \int 2 sin (v) + 5 d v

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

= \int 2 sin (v) d v + \int 5 d v

\int 2 sin (v) d v = - 2 cos (v)

\int 5 d v = 5 v

= - 2 cos (v) + 5 v

代回

= - 2 cos (arcsin (\frac{1}{2} (x - 2))) + 5 arcsin (\frac{1}{2} (x - 2))

化简

- 2 cos (arcsin (\frac{1}{2} (x - 2))) + 5 arcsin (\frac{1}{2} (x - 2)) : - - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1)

= - - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1)

解答补常数

= - - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1) + C

d er i v a t i v e 2^{90}

d er i v a t i v e f (x) = 2 x^{2} - 7 x + 5

\frac{d}{d x} (- 2 x sin (2 x))

d er i v a t i v e f (t) = cos (t)

\frac{\partial}{\partial x} (x^{2} y^{3} + 4 x yz - 3 yz)