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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用三角换元法

= \int 2 (sin (u) + 1) d u

提出常数:

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

= 2 \cdot \int sin (u) + 1 d u

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

= 2 (\int sin (u) d u + \int 1 d u)

\int sin (u) d u = - cos (u)

\int 1 d u = u

= 2 (- cos (u) + u)

u = arcsin (\frac{1}{2} x)

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of sec^2(x/6)

\int sec^{2} (\frac{x}{6}) d x

integral of sin^4(x/9)cos(x/9)

\int sin^{4} (\frac{x}{9}) cos (\frac{x}{9}) d x

积分 1/((x^2+1))

in t e g r a l \frac{1}{( x ^{2} + 1 )}

integral of 4/(sqrt(1-4x^2))

\int \frac{4}{1 - 4 x ^{2}} d x

integral of csc^2(x+1)

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