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受欢迎的微积分 >

integral of 1/(1+sqrt(x-1))

解答

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

解答

2 (1 + x - 1 - ln 1 + x - 1) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

= 2 \cdot \int \frac{u}{1 + u} d u

使用换元积分法

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

乘开

\frac{v - 1}{v} : 1 - \frac{1}{v}

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

= 2 (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 2 (v - ln ∣ v ∣)

代回

= 2 (1 + x - 1 - ln 1 + x - 1)

解答补常数

= 2 (1 + x - 1 - ln 1 + x - 1) + C

作图

流行的例子

integral of 1/((x^{2/3))(1+x^{2/3})}

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

integral of (4x^2)/(sqrt(1+x^3))

\int \frac{4 x ^{2}}{1 + x ^{3}} d x

integral of 6x-3x^2

\int 6 x - 3 x^{2} d x

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

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

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

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