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人気のある微分積分 >

integral of 1/(4x^2-9)

解

\int \frac{1}{4 x ^{2} - 9} d x

解

- \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1) + C

解答ステップ

\int \frac{1}{4 x ^{2} - 9} d x

置換積分法を適用する

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

定数を除く:

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

= \frac{1}{6} \cdot \int \frac{1}{u ^{2} - 1} d u

因数

u^{2} - 1 : - (- u^{2} + 1)

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

定数を除く:

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

= \frac{1}{6} (- \int \frac{1}{- u ^{2} + 1} d u)

共通積分を使用する:

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

= \frac{1}{6} (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

代用を戻す

u = \frac{2}{3} x

= \frac{1}{6} (- (\frac{ln \frac{2}{3} x + 1}{2} - \frac{ln \frac{2}{3} x - 1}{2}))

簡素化

\frac{1}{6} (- (\frac{ln \frac{2}{3} x + 1}{2} - \frac{ln \frac{2}{3} x - 1}{2})) : - \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1)

= - \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1)

定数を解答に追加する

= - \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1) + C

グラフ

人気の例

derivative of 3sin(xcos(x))

\frac{d}{d x} (3 sin (x) cos (x))

derivative f(x)=x^{-1}-1/x

d er i v a t i v e f (x) = x^{- 1} - \frac{1}{x}

derivative of 2/(3x-1)

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

integral of y^2cos(x)+z^3

\int y^{2} cos (x) + z^{3} d x

f^{''}(x)=4x^2+8x-17

f^{^{''}} (x) = 4 x^{2} + 8 x - 17