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受欢迎的微积分 >

积分 sqrt(((x-1))/((x+1)))

解答

积分

\frac{( x - 1 )}{( x + 1 )}

解答

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

求解步骤

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

化简

\frac{x - 1}{x + 1} : \frac{x - 1}{x + 1}

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

使用换元积分法

= \int 2 u^{2} - 2 d u

提出常数:

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

= 2 \cdot \int u^{2} - 2 d u

使用三角换元法

= 2 \cdot \int 2 tan^{2} (v) sec (v) d v

提出常数:

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

= 2 \cdot 2 \cdot \int tan^{2} (v) sec (v) d v

使用三角恒等式改写

= 2 \cdot 2 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

乘开

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 2 \cdot 2 \cdot \int - sec (v) + sec^{3} (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 \cdot 2 (- \int sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 2 \cdot 2 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 2 \cdot 2 (- ln tan (arcsec (\frac{1}{2} x + 1)) + sec (arcsec (\frac{1}{2} x + 1)) + \frac{1}{2} sec (arcsec (\frac{1}{2} x + 1)) tan (arcsec (\frac{1}{2} x + 1)) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} x + 1)) + sec (arcsec (\frac{1}{2} x + 1)))

化简

2 \cdot 2 (- ln tan (arcsec (\frac{1}{2} x + 1)) + sec (arcsec (\frac{1}{2} x + 1)) + \frac{1}{2} sec (arcsec (\frac{1}{2} x + 1)) tan (arcsec (\frac{1}{2} x + 1)) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} x + 1)) + sec (arcsec (\frac{1}{2} x + 1))) : - 2 ln \frac{1}{2} (x - 1) + \frac{1}{2} x + 1 + x - 1 x + 1

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

解答补常数

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

作图

流行的例子

f(x)= x/((x^2+x+1)^{1/2)}

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

积分 t/((t+2))

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

积分 (tan(x))/((1+3tan^2(x)))

in t e g r a l \frac{tan ( x )}{( 1 + 3 tan ^{2} ( x ) )}

积分 log_{10}(a^2+x^2)

in t e g r a l lo g_{10} (a^{2} + x^{2})

积分 (x^3)/((x^2+a^2))

in t e g r a l \frac{x ^{3}}{( x ^{2} + a ^{2} )}