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人気のある微分積分 >

integral of 1/(xsqrt(12x+25))

解

\int \frac{1}{x 12 x + 25} d x

解

- \frac{1}{5} (ln \frac{12 x + 25}{5} + 1 - ln \frac{12 x + 25}{5} - 1) + C

解答ステップ

\int \frac{1}{x 12 x + 25} d x

u置換積分法を適用する

= \int \frac{2}{u ^{2} - 25} d u

定数を除く:

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

= 2 \cdot \int \frac{1}{u ^{2} - 25} d u

置換積分法を適用する

= 2 \cdot \int \frac{1}{5 ( v ^{2} - 1 )} d v

定数を除く:

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

= 2 \cdot \frac{1}{5} \cdot \int \frac{1}{v ^{2} - 1} d v

因数

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

= 2 \cdot \frac{1}{5} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

定数を除く:

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

= 2 \cdot \frac{1}{5} (- \int \frac{1}{- v ^{2} + 1} d v)

共通積分を使用する:

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

= 2 \cdot \frac{1}{5} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

代用を戻す

= 2 \cdot \frac{1}{5} - \frac{ln \frac{12 x + 25}{5} + 1}{2} - \frac{ln \frac{12 x + 25}{5} - 1}{2}

簡素化

2 \cdot \frac{1}{5} - \frac{ln \frac{12 x + 25}{5} + 1}{2} - \frac{ln \frac{12 x + 25}{5} - 1}{2} : - \frac{1}{5} (ln \frac{12 x + 25}{5} + 1 - ln \frac{12 x + 25}{5} - 1)

= - \frac{1}{5} (ln \frac{12 x + 25}{5} + 1 - ln \frac{12 x + 25}{5} - 1)

定数を解答に追加する

= - \frac{1}{5} (ln \frac{12 x + 25}{5} + 1 - ln \frac{12 x + 25}{5} - 1) + C

グラフ

人気の例

limit as x approaches 0-of sqrt(-x)

x \to 0 - lim (- x)

derivative of 2-y^2+x^2

\frac{d}{d x} (2 - y^{2} + x^{2})

slopeintercept (1,1),(-4,-3)

s l o p e in t erce pt (1, 1), (- 4, - 3)

y^{^{'}} = - 4 x y^{3}

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