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受欢迎的微积分 >

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

解答

\int \frac{1}{t - 1 - t ^{2}} d t

解答

\frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 2 tan (\frac{1}{2} arcsin (t)) - 1 - \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 1 - arctan (tan (\frac{1}{2} arcsin (t))) + C

求解步骤

\int \frac{1}{t - 1 - t ^{2}} d t

使用三角换元法

= \int \frac{cos ( u )}{sin ( u ) - cos ( u )} d u

使用换元积分法

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

提出常数:

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

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

将

\frac{1 - v ^{2}}{( 2 v - 1 + v ^{2} ) ( 1 + v ^{2} )}

用部份分式展开:

\frac{v + 1}{2 ( v ^{2} + 2 v - 1 )} + \frac{- v - 1}{2 ( v ^{2} + 1 )}

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

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

\int \frac{- v - 1}{2 ( v ^{2} + 1 )} d v = \frac{1}{2} (- \frac{1}{2} ln v^{2} + 1 - arctan (v))

= 2 (\frac{1}{4} ln v^{2} + 2 v - 1 + \frac{1}{2} (- \frac{1}{2} ln v^{2} + 1 - arctan (v)))

代回

= 2 (\frac{1}{4} ln tan^{2} (\frac{arcsin ( t )}{2}) + 2 tan (\frac{arcsin ( t )}{2}) - 1 + \frac{1}{2} (- \frac{1}{2} ln tan^{2} (\frac{arcsin ( t )}{2}) + 1 - arctan (tan (\frac{arcsin ( t )}{2}))))

化简

2 (\frac{1}{4} ln tan^{2} (\frac{arcsin ( t )}{2}) + 2 tan (\frac{arcsin ( t )}{2}) - 1 + \frac{1}{2} (- \frac{1}{2} ln tan^{2} (\frac{arcsin ( t )}{2}) + 1 - arctan (tan (\frac{arcsin ( t )}{2})))) : \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 2 tan (\frac{1}{2} arcsin (t)) - 1 - \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 1 - arctan (tan (\frac{1}{2} arcsin (t)))

= \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 2 tan (\frac{1}{2} arcsin (t)) - 1 - \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 1 - arctan (tan (\frac{1}{2} arcsin (t)))

解答补常数

= \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 2 tan (\frac{1}{2} arcsin (t)) - 1 - \frac{1}{2} ln tan^{2} (\frac{1}{2} arcsin (t)) + 1 - arctan (tan (\frac{1}{2} arcsin (t))) + C

作图

流行的例子

integral of 21sin^4(x)

\int 21 sin^{4} (x) d x

integral of (2u)/(1-u)

\int \frac{2 u}{1 - u} d u

integral of ((ln(53))/v-ln(6)v^e)

\int (\frac{ln ( 53 )}{v} - ln (6) v^{e}) d v

integral of (arcsin(x))/(3x^2)

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

integral of (tan(x)+sec(x))/(sec^2(x))

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