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受欢迎的微积分 >

integral of 1/(sqrt(3+x-x^2))

解答

\int \frac{1}{3 + x - x ^{2}} d x

解答

arcsin (\frac{1}{13} (2 x - 1)) + C

求解步骤

\int \frac{1}{3 + x - x ^{2}} d x

对

3 + x - x^{2}

配方:

- (x - \frac{1}{2})^{2} + \frac{13}{4}

= \int \frac{1}{- ( x - \frac{1}{2} ) ^{2} + \frac{13}{4}} d x

使用换元积分法

= \int \frac{2}{- 4 u ^{2} + 13} d u

提出常数:

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

= 2 \cdot \int \frac{1}{- 4 u ^{2} + 13} d u

使用三角换元法

= 2 \cdot \int \frac{1}{2} d v

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int 1 d v

常数积分:

\int a d x = a x

= 2 \cdot \frac{1}{2} \cdot 1 \cdot v

代回

= 2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{13} (x - \frac{1}{2}))

化简

2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{13} (x - \frac{1}{2})) : arcsin (\frac{1}{13} (2 x - 1))

= arcsin (\frac{1}{13} (2 x - 1))

解答补常数

= arcsin (\frac{1}{13} (2 x - 1)) + C

作图

流行的例子

derivative (x^3-2x+1)(3x^3+2x^2-5x)

d er i v a t i v e (x^{3} - 2 x + 1) (3 x^{3} + 2 x^{2} - 5 x)

derivative y=((x+1)^2)/(sqrt(x)+1)

d er i v a t i v e y = \frac{( x + 1 ) ^{2}}{x + 1}

((3y^2-t^2))/(y^5)dy+t/(2y^4)dt=0

\frac{( 3 y ^{2} - t ^{2} )}{y ^{5}} d y + \frac{t}{2 y ^{4}} d t = 0

sum from n=0 to infinity of (3/5)^n

n = 0 \sum \infty (\frac{3}{5})^{n}

limit as x approaches+3-of 3/(x-3)

x \to + 3 - lim (\frac{3}{x - 3})