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인기 있는 미적분학 >

적분 (-4x)/((x^4-2x^2+1))

해법

적분

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

해법

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

솔루션 단계

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

정수를 빼라:

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

= - 4 \cdot \int \frac{x}{x ^{4} - 2 x ^{2} + 1} d x

부품별 통합 적용

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

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

= - 4 (x (\frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )} - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{4 ( x - 1 )}) - \frac{x ( x ^{2} ln ∣ x + 1 ∣ - x ^{2} ln ∣ x - 1 ∣ - 2 x + ln ∣ x - 1 ∣ - ln ∣ x + 1 ∣ ) - ( \frac{1}{x + 1} - \frac{1}{x - 1} ) ( x + 1 ) ( x - 1 )}{4 ( x + 1 ) ( x - 1 )})

- 4 (x (\frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )} - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{4 ( x - 1 )}) - \frac{x ( x ^{2} ln ∣ x + 1 ∣ - x ^{2} ln ∣ x - 1 ∣ - 2 x + ln ∣ x - 1 ∣ - ln ∣ x + 1 ∣ ) - ( \frac{1}{x + 1} - \frac{1}{x - 1} ) ( x + 1 ) ( x - 1 )}{4 ( x + 1 ) ( x - 1 )})

간소화하다 :

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

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

솔루션에 상수 추가

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

그래프

인기 있는 예

in t e g r a l \frac{5}{x ^{3}}

적분 x^2sec^2(x)

in t e g r a l x^{2} sec^{2} (x)

적분 1/((100+x^2)^2)

in t e g r a l \frac{1}{( 100 + x ^{2} ) ^{2}}

in t e g r a l \frac{- 2}{t}

적분 (5x-4)^3

in t e g r a l (5 x - 4)^{3}