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受欢迎的 >

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

解答

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

解答

3 x^{2} + 2 x + 2 - 4 ln x^{2} + 2 x + 2 + x + 1 + C

求解步骤

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

对

x^{2} + 2 x + 2

配方:

(x + 1)^{2} + 1

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

使用换元积分法

: \int \frac{3 u - 4}{u ^{2} + 1} d u

= \int \frac{3 u - 4}{u ^{2} + 1} d u

乘开

\frac{3 u - 4}{u ^{2} + 1} : \frac{3 u}{u ^{2} + 1} - \frac{4}{u ^{2} + 1}

= \int \frac{3 u}{u ^{2} + 1} - \frac{4}{u ^{2} + 1} 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

= \int \frac{3 u}{u ^{2} + 1} d u - \int \frac{4}{u ^{2} + 1} d u

\int \frac{3 u}{u ^{2} + 1} d u = 3 u^{2} + 1

\int \frac{4}{u ^{2} + 1} d u = 4 ln u^{2} + 1 + u

= 3 u^{2} + 1 - 4 ln u^{2} + 1 + u

u = x + 1

代回

= 3 (x + 1)^{2} + 1 - 4 ln (x + 1)^{2} + 1 + x + 1

化简

3 (x + 1)^{2} + 1 - 4 ln (x + 1)^{2} + 1 + x + 1 : 3 x^{2} + 2 x + 2 - 4 ln x^{2} + 2 x + 2 + x + 1

= 3 x^{2} + 2 x + 2 - 4 ln x^{2} + 2 x + 2 + x + 1

解答补常数

= 3 x^{2} + 2 x + 2 - 4 ln x^{2} + 2 x + 2 + x + 1 + C

流行的例子

integral of (7cos(x))/(2sin(x)+1)

\int \frac{7 cos ( x )}{2 sin ( x ) + 1} d x

integral of (x^2+2)/(4x^5+4x^3+x)

\int \frac{x ^{2} + 2}{4 x ^{5} + 4 x ^{3} + x} d x

integral of (3x+1)/(x+1)

\int \frac{3 x + 1}{x + 1} d x

integral of (-8x+16)/((x^2+1)(x-1))

\int \frac{- 8 x + 16}{( x ^{2} + 1 ) ( x - 1 )} d x

integral of (x[ln(1+x)+ln(1-x)])/(x^2-1)

\int \frac{x [ ln ( 1 + x ) + ln ( 1 - x ) ]}{x ^{2} - 1} d x