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受欢迎的微积分 >

integral of 1/(sqrt(x)+\sqrt{x+1)}

解答

\int \frac{1}{x + x + 1} d x

解答

\frac{3 x ( x + x + 1 ) ^{3} + 3 ( x + x + 1 ) ^{3} x + 1 + 1}{6 ( x + x + 1 ) ^{3}} + C

求解步骤

\int \frac{1}{x + x + 1} d x

使用换元积分法

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

提出常数:

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

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

乘开

\frac{( u + 1 ) ( u - 1 ) ( u ^{2} + 1 )}{u ^{4}} : 1 - \frac{1}{u ^{4}}

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

= \frac{1}{2} (\int 1 d u - \int \frac{1}{u ^{4}} d u)

\int 1 d u = u

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

= \frac{1}{2} (u - (- \frac{1}{3 u ^{3}}))

u = x + x + 1

代回

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

化简

\frac{1}{2} (x + x + 1 - (- \frac{1}{3 ( x + x + 1 ) ^{3}})) : \frac{3 x ( x + x + 1 ) ^{3} + 3 ( x + x + 1 ) ^{3} x + 1 + 1}{6 ( x + x + 1 ) ^{3}}

= \frac{3 x ( x + x + 1 ) ^{3} + 3 ( x + x + 1 ) ^{3} x + 1 + 1}{6 ( x + x + 1 ) ^{3}}

解答补常数

= \frac{3 x ( x + x + 1 ) ^{3} + 3 ( x + x + 1 ) ^{3} x + 1 + 1}{6 ( x + x + 1 ) ^{3}} + C

作图

流行的例子

(dy)/(dt)=e^{-3t}

\frac{d y}{d t} = e^{- 3 t}

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p a r i t y cos^{x} (x)

(d^2)/(dx^2)((2x)/(x^2+1))

\frac{d ^{2}}{d x ^{2}} (\frac{2 x}{x ^{2} + 1})

derivative of 1/(2x^2-3x+1)

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

sum from n=1 to infinity of (n^5)/(5^n)

n = 1 \sum \infty \frac{n ^{5}}{5 ^{n}}