integral of 2x(x^2+1)^{-7}

해법

\int 2 x (x^{2} + 1)^{- 7} d x

- \frac{1}{6 ( x ^{2} + 1 ) ^{6}} + C

솔루션 단계

\int 2 x (x^{2} + 1)^{- 7} d x

정수를 빼라:

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

= 2 \cdot \int x (x^{2} + 1)^{- 7} d x

x (x^{2} + 1)^{- 7}

단순화하세요:

\frac{x}{( x ^{2} + 1 ) ^{7}}

= 2 \cdot \int \frac{x}{( x ^{2} + 1 ) ^{7}} d x

대체 적용

: \int \frac{1}{2 u ^{7}} d u

= 2 \cdot \int \frac{1}{2 u ^{7}} d u

정수를 빼라:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{7}} d u

전원 규칙 적용

: - \frac{1}{6 u ^{6}}

= 2 \cdot \frac{1}{2} (- \frac{1}{6 u ^{6}})

뒤로 대체

u = x^{2} + 1

= 2 \cdot \frac{1}{2} (- \frac{1}{6 ( x ^{2} + 1 ) ^{6}})

2 \cdot \frac{1}{2} (- \frac{1}{6 ( x ^{2} + 1 ) ^{6}})

간소화하다 :

- \frac{1}{6 ( x ^{2} + 1 ) ^{6}}

= - \frac{1}{6 ( x ^{2} + 1 ) ^{6}}

솔루션에 상수 추가

= - \frac{1}{6 ( x ^{2} + 1 ) ^{6}} + C